Results of Zeta and Related Functions

DLMF	Formula	Maple	Mathematica	Symbolic Maple	Symbolic Mathematica	Numeric Maple	Numeric Mathematica
25.2.E1	${\displaystyle{\displaystyle\zeta\left(s\right)=\sum_{n=1}^{\infty}\frac{1}{n^{s}}}}$	Zeta(s)= sum((1)/((n)^(s)), n = 1..infinity)	Zeta[s]= Sum[Divide[1,(n)^(s)], {n, 1, Infinity}]	Failure	Successful	Skip	-
25.2.E2	${\displaystyle{\displaystyle\zeta\left(s\right)=\frac{1}{1-2^{-s}}\sum_{n=0}^{\infty}\frac{1}{(2n+1)^{s}}}}$	Zeta(s)=(1)/(1 - (2)^(- s))*sum((1)/((2*n + 1)^(s)), n = 0..infinity)	Zeta[s]=Divide[1,1 - (2)^(- s)]*Sum[Divide[1,(2*n + 1)^(s)], {n, 0, Infinity}]	Successful	Successful	-	-
25.2.E3	${\displaystyle{\displaystyle\zeta\left(s\right)=\frac{1}{1-2^{1-s}}\sum_{n=1}^{\infty}\frac{(-1)^{n-1}}{n^{s}}}}$	Zeta(s)=(1)/(1 - (2)^(1 - s))*sum(((- 1)^(n - 1))/((n)^(s)), n = 1..infinity)	Zeta[s]=Divide[1,1 - (2)^(1 - s)]*Sum[Divide[(- 1)^(n - 1),(n)^(s)], {n, 1, Infinity}]	Failure	Successful	Skip	-
25.2.E4	${\displaystyle{\displaystyle\zeta\left(s\right)=\frac{1}{s-1}+\sum_{n=0}^{\infty}\frac{(-1)^{n}}{n!}\gamma_{n}(s-1)^{n}}}$	Zeta(s)=(1)/(s - 1)+ sum(((- 1)^(n))/(factorial(n))*gamma[n]*(s - 1)^(n), n = 0..infinity)	Zeta[s]=Divide[1,s - 1]+ Sum[Divide[(- 1)^(n),(n)!]*Subscript[\[Gamma], n]*(s - 1)^(n), {n, 0, Infinity}]	Failure	Failure	Skip	Skip
25.2.E6	${\displaystyle{\displaystyle\zeta'\left(s\right)=-\sum_{n=2}^{\infty}(\ln n)n^{-s}}}$	subs( temp=s, diff( Zeta(temp), temp$(1) ) )= - sum((ln(n))* (n)^(- s), n = 2..infinity)	(D[Zeta[temp], {temp, 1}]/.temp-> s)= - Sum[(Log[n])* (n)^(- s), {n, 2, Infinity}]	Successful	Successful	-	-
25.2.E7	${\displaystyle{\displaystyle{\zeta^{(k)}}\left(s\right)=(-1)^{k}\sum_{n=2}^{\infty}(\ln n)^{k}n^{-s}}}$	subs( temp=s, diff( Zeta(temp), temp$(k) ) )=(- 1)^(k)* sum((ln(n))^(k)* (n)^(- s), n = 2..infinity)	(D[Zeta[temp], {temp, k}]/.temp-> s)=(- 1)^(k)* Sum[(Log[n])^(k)* (n)^(- s), {n, 2, Infinity}]	Failure	Failure	Skip	Successful
25.2.E8	${\displaystyle{\displaystyle\zeta\left(s\right)=\sum_{k=1}^{N}\frac{1}{k^{s}}+\frac{N^{1-s}}{s-1}-s\int_{N}^{\infty}\frac{x-\left\lfloor x\right\rfloor}{x^{s+1}}\mathrm{d}x}}$	Zeta(s)= sum((1)/((k)^(s)), k = 1..N)+((N)^(1 - s))/(s - 1)- s*int((x - floor(x))/((x)^(s + 1)), x = N..infinity)	Zeta[s]= Sum[Divide[1,(k)^(s)], {k, 1, N}]+Divide[(N)^(1 - s),s - 1]- s*Integrate[Divide[x - Floor[x],(x)^(s + 1)], {x, N, Infinity}]	Failure	Failure	Skip	Successful
25.2.E11	${\displaystyle{\displaystyle\zeta\left(s\right)=\prod_{p}(1-p^{-s})^{-1}}}$	Zeta(s)= product((1 - (p)^(- s))^(- 1), p = - infinity..infinity)	Zeta[s]= Product[(1 - (p)^(- s))^(- 1), {p, - Infinity, Infinity}]	Failure	Failure	Skip	-
25.2.E12	${\displaystyle{\displaystyle\zeta\left(s\right)=\frac{(2\pi)^{s}e^{-s-(\gamma s/2)}}{2(s-1)\Gamma\left(\tfrac{1}{2}s+1\right)}\prod_{\rho}\left(1-\frac{s}{\rho}\right)e^{s/\rho}}}$	Zeta(s)=((2*Pi)^(s)* exp(- s -(gamma*s/ 2)))/(2*(s - 1)* GAMMA((1)/(2)*s + 1))*product((1 -(s)/(rho))* exp(s/ rho), rho = - infinity..infinity)	Zeta[s]=Divide[(2*Pi)^(s)* Exp[- s -(EulerGamma*s/ 2)],2*(s - 1)* Gamma[Divide[1,2]*s + 1]]*Product[(1 -Divide[s,\[Rho]])* Exp[s/ \[Rho]], {\[Rho], - Infinity, Infinity}]	Failure	Failure	Skip	Skip
25.4.E1	${\displaystyle{\displaystyle\zeta\left(1-s\right)=2(2\pi)^{-s}\cos\left(\tfrac{1}{2}\pi s\right)\Gamma\left(s\right)\zeta\left(s\right)}}$	Zeta(1 - s)= 2*(2*Pi)^(- s)* cos((1)/(2)*Pi*s)*GAMMA(s)*Zeta(s)	Zeta[1 - s]= 2*(2*Pi)^(- s)* Cos[Divide[1,2]*Pi*s]*Gamma[s]*Zeta[s]	Failure	Successful	Successful	-
25.4.E2	${\displaystyle{\displaystyle\zeta\left(s\right)=2(2\pi)^{s-1}\sin\left(\tfrac{1}{2}\pi s\right)\Gamma\left(1-s\right)\zeta\left(1-s\right)}}$	Zeta(s)= 2*(2*Pi)^(s - 1)* sin((1)/(2)*Pi*s)*GAMMA(1 - s)*Zeta(1 - s)	Zeta[s]= 2*(2*Pi)^(s - 1)* Sin[Divide[1,2]*Pi*s]*Gamma[1 - s]*Zeta[1 - s]	Failure	Successful	Successful	-
25.4.E3	${\displaystyle{\displaystyle\xi\left(s\right)=\xi\left(1-s\right)}}$	(s)*(s-1)*GAMMA((s)/2)*Pi^(-(s)/2)*Zeta(s)/2 = (1 - s)*(1 - s-1)*GAMMA((1 - s)/2)*Pi^(-(1 - s)/2)*Zeta(1 - s)/2	Error	Failure	Error	Successful	-
25.4.E4	${\displaystyle{\displaystyle\xi\left(s\right)=\tfrac{1}{2}s(s-1)\Gamma\left(\tfrac{1}{2}s\right)\pi^{-s/2}\zeta\left(s\right)}}$	(s)*(s-1)*GAMMA((s)/2)*Pi^(-(s)/2)*Zeta(s)/2 =(1)/(2)*s*(s - 1)* GAMMA((1)/(2)*s)*(Pi)^(- s/ 2)* Zeta(s)	Error	Successful	Error	-	-
25.4.E5	${\displaystyle{\displaystyle(-1)^{k}{\zeta^{(k)}}\left(1-s\right)=\frac{2}{(2\pi)^{s}}\sum_{m=0}^{k}\sum_{r=0}^{m}\genfrac{(}{)}{0.0pt}{}{k}{m}\genfrac{(}{)}{0.0pt}{}{m}{r}\left(\Re(c^{k-m})\cos\left(\tfrac{1}{2}\pi s\right)+\Im(c^{k-m})\sin\left(\tfrac{1}{2}\pi s\right)\right){\Gamma^{(r)}}\left(s\right){\zeta^{(m-r)}}\left(s\right)}}$	(- 1)^(k)* subs( temp=1 - s, diff( Zeta(temp), temp$(k) ) )=(2)/((2*Pi)^(s))*sum(sum(binomial(k,m)*binomial(m,r)*(Re((c)^(k - m))*cos((1)/(2)*Pi*s)+ Im((c)^(k - m))*sin((1)/(2)*Pi*s))* subs( temp=s, diff( GAMMA(temp), temp$(r) ) )*subs( temp=s, diff( Zeta(temp), temp$(m - r) ) ), r = 0..m), m = 0..k)	(- 1)^(k)* (D[Zeta[temp], {temp, k}]/.temp-> 1 - s)=Divide[2,(2*Pi)^(s)]*Sum[Sum[Binomial[k,m]*Binomial[m,r]*(Re[(c)^(k - m)]*Cos[Divide[1,2]*Pi*s]+ Im[(c)^(k - m)]*Sin[Divide[1,2]*Pi*s])* (D[Gamma[temp], {temp, r}]/.temp-> s)*(D[Zeta[temp], {temp, m - r}]/.temp-> s), {r, 0, m}], {m, 0, k}]	Failure	Failure	Skip	Skip
25.4.E6	${\displaystyle{\displaystyle c=-\ln\left(2\pi\right)-\tfrac{1}{2}\pi\mathrm{i}}}$	c = - ln(2*Pi)-(1)/(2)*Pi*I	c = - Log[2*Pi]-Divide[1,2]*Pi*I	Failure	Failure	Fail 3.252090629+2.985009889*I <- {c = 2^(1/2)+I*2^(1/2)} 3.252090629+.156582765*I <- {c = 2^(1/2)-I*2^(1/2)} .423663505+.156582765*I <- {c = -2^(1/2)-I*2^(1/2)} .423663505+2.985009889*I <- {c = -2^(1/2)+I*2^(1/2)}	Fail Complex[3.2520906287824403, 2.9850098891679915] <- {Rule[c, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]} Complex[3.2520906287824403, 0.1565827644218014] <- {Rule[c, Times[Complex[1, -1], Power[2, Rational[1, 2]]]]} Complex[0.4236635040362502, 0.1565827644218014] <- {Rule[c, Times[Complex[-1, -1], Power[2, Rational[1, 2]]]]} Complex[0.4236635040362502, 2.9850098891679915] <- {Rule[c, Times[Complex[-1, 1], Power[2, Rational[1, 2]]]]}
25.5.E1	${\displaystyle{\displaystyle\zeta\left(s\right)=\frac{1}{\Gamma\left(s\right)}\int_{0}^{\infty}\frac{x^{s-1}}{e^{x}-1}\mathrm{d}x}}$	Zeta(s)=(1)/(GAMMA(s))*int(((x)^(s - 1))/(exp(x)- 1), x = 0..infinity)	Zeta[s]=Divide[1,Gamma[s]]*Integrate[Divide[(x)^(s - 1),Exp[x]- 1], {x, 0, Infinity}]	Failure	Successful	Skip	-
25.5.E2	${\displaystyle{\displaystyle\zeta\left(s\right)=\frac{1}{\Gamma\left(s+1\right)}\int_{0}^{\infty}\frac{e^{x}x^{s}}{(e^{x}-1)^{2}}\mathrm{d}x}}$	Zeta(s)=(1)/(GAMMA(s + 1))*int((exp(x)*(x)^(s))/((exp(x)- 1)^(2)), x = 0..infinity)	Zeta[s]=Divide[1,Gamma[s + 1]]*Integrate[Divide[Exp[x]*(x)^(s),(Exp[x]- 1)^(2)], {x, 0, Infinity}]	Failure	Failure	Skip	Skip
25.5.E3	${\displaystyle{\displaystyle\zeta\left(s\right)=\frac{1}{(1-2^{1-s})\Gamma\left(s\right)}\int_{0}^{\infty}\frac{x^{s-1}}{e^{x}+1}\mathrm{d}x}}$	Zeta(s)=(1)/((1 - (2)^(1 - s))* GAMMA(s))*int(((x)^(s - 1))/(exp(x)+ 1), x = 0..infinity)	Zeta[s]=Divide[1,(1 - (2)^(1 - s))* Gamma[s]]*Integrate[Divide[(x)^(s - 1),Exp[x]+ 1], {x, 0, Infinity}]	Failure	Successful	Skip	-
25.5.E4	${\displaystyle{\displaystyle\zeta\left(s\right)=\frac{1}{(1-2^{1-s})\Gamma\left(s+1\right)}\int_{0}^{\infty}\frac{e^{x}x^{s}}{(e^{x}+1)^{2}}\mathrm{d}x}}$	Zeta(s)=(1)/((1 - (2)^(1 - s))* GAMMA(s + 1))*int((exp(x)*(x)^(s))/((exp(x)+ 1)^(2)), x = 0..infinity)	Zeta[s]=Divide[1,(1 - (2)^(1 - s))* Gamma[s + 1]]*Integrate[Divide[Exp[x]*(x)^(s),(Exp[x]+ 1)^(2)], {x, 0, Infinity}]	Failure	Failure	Skip	Skip
25.5.E5	${\displaystyle{\displaystyle\zeta\left(s\right)=-s\int_{0}^{\infty}\frac{x-\left\lfloor x\right\rfloor-\frac{1}{2}}{x^{s+1}}\mathrm{d}x}}$	Zeta(s)= - s*int((x - floor(x)-(1)/(2))/((x)^(s + 1)), x = 0..infinity)	Zeta[s]= - s*Integrate[Divide[x - Floor[x]-Divide[1,2],(x)^(s + 1)], {x, 0, Infinity}]	Failure	Failure	Skip	Successful
25.5.E6	${\displaystyle{\displaystyle\zeta\left(s\right)=\frac{1}{2}+\frac{1}{s-1}+\frac{1}{\Gamma\left(s\right)}\int_{0}^{\infty}\left(\frac{1}{e^{x}-1}-\frac{1}{x}+\frac{1}{2}\right)\frac{x^{s-1}}{e^{x}}\mathrm{d}x}}$	Zeta(s)=(1)/(2)+(1)/(s - 1)+(1)/(GAMMA(s))*int(((1)/(exp(x)- 1)-(1)/(x)+(1)/(2))*((x)^(s - 1))/(exp(x)), x = 0..infinity)	Zeta[s]=Divide[1,2]+Divide[1,s - 1]+Divide[1,Gamma[s]]*Integrate[(Divide[1,Exp[x]- 1]-Divide[1,x]+Divide[1,2])*Divide[(x)^(s - 1),Exp[x]], {x, 0, Infinity}]	Failure	Failure	Skip	Successful
25.5.E7	${\displaystyle{\displaystyle\zeta\left(s\right)=\frac{1}{2}+\frac{1}{s-1}+\sum_{m=1}^{n}\frac{B_{2m}}{(2m)!}{\left(s\right)_{2m-1}}+\frac{1}{\Gamma\left(s\right)}\int_{0}^{\infty}\left(\frac{1}{e^{x}-1}-\frac{1}{x}+\frac{1}{2}-\sum_{m=1}^{n}\frac{B_{2m}}{(2m)!}x^{2m-1}\right)\frac{x^{s-1}}{e^{x}}\mathrm{d}x}}$	Zeta(s)=(1)/(2)+(1)/(s - 1)+ sum((bernoulli(2*m))/(factorial(2*m))*pochhammer(s, 2*m - 1)+(1)/(GAMMA(s))*int(((1)/(exp(x)- 1)-(1)/(x)+(1)/(2)- sum((bernoulli(2*m))/(factorial(2*m))*(x)^(2*m - 1), m = 1..n))*((x)^(s - 1))/(exp(x)), x = 0..infinity), m = 1..n)	Zeta[s]=Divide[1,2]+Divide[1,s - 1]+ Sum[Divide[BernoulliB[2*m],(2*m)!]*Pochhammer[s, 2*m - 1]+Divide[1,Gamma[s]]*Integrate[(Divide[1,Exp[x]- 1]-Divide[1,x]+Divide[1,2]- Sum[Divide[BernoulliB[2*m],(2*m)!]*(x)^(2*m - 1), {m, 1, n}])*Divide[(x)^(s - 1),Exp[x]], {x, 0, Infinity}], {m, 1, n}]	Failure	Failure	Skip	Error
25.5.E8	${\displaystyle{\displaystyle\zeta\left(s\right)=\frac{1}{2(1-2^{-s})\Gamma\left(s\right)}\int_{0}^{\infty}\frac{x^{s-1}}{\sinh x}\mathrm{d}x}}$	Zeta(s)=(1)/(2*(1 - (2)^(- s))* GAMMA(s))*int(((x)^(s - 1))/(sinh(x)), x = 0..infinity)	Zeta[s]=Divide[1,2*(1 - (2)^(- s))* Gamma[s]]*Integrate[Divide[(x)^(s - 1),Sinh[x]], {x, 0, Infinity}]	Failure	Failure	Skip	Skip
25.5.E9	${\displaystyle{\displaystyle\zeta\left(s\right)=\frac{2^{s-1}}{\Gamma\left(s+1\right)}\int_{0}^{\infty}\frac{x^{s}}{(\sinh x)^{2}}\mathrm{d}x}}$	Zeta(s)=((2)^(s - 1))/(GAMMA(s + 1))*int(((x)^(s))/((sinh(x))^(2)), x = 0..infinity)	Zeta[s]=Divide[(2)^(s - 1),Gamma[s + 1]]*Integrate[Divide[(x)^(s),(Sinh[x])^(2)], {x, 0, Infinity}]	Failure	Failure	Skip	-
25.5.E10	${\displaystyle{\displaystyle\zeta\left(s\right)=\frac{2^{s-1}}{1-2^{1-s}}\int_{0}^{\infty}\frac{\cos\left(s\operatorname{arctan}x\right)}{(1+x^{2})^{s/2}\cosh\left(\frac{1}{2}\pi x\right)}\mathrm{d}x}}$	Zeta(s)=((2)^(s - 1))/(1 - (2)^(1 - s))*int((cos(s*arctan(x)))/((1 + (x)^(2))^(s/ 2)* cosh((1)/(2)*Pi*x)), x = 0..infinity)	Zeta[s]=Divide[(2)^(s - 1),1 - (2)^(1 - s)]*Integrate[Divide[Cos[s*ArcTan[x]],(1 + (x)^(2))^(s/ 2)* Cosh[Divide[1,2]*Pi*x]], {x, 0, Infinity}]	Failure	Failure	Skip	Error
25.5.E11	${\displaystyle{\displaystyle\zeta\left(s\right)=\frac{1}{2}+\frac{1}{s-1}+2\int_{0}^{\infty}\frac{\sin\left(s\operatorname{arctan}x\right)}{(1+x^{2})^{s/2}(e^{2\pi x}-1)}\mathrm{d}x}}$	Zeta(s)=(1)/(2)+(1)/(s - 1)+ 2*int((sin(s*arctan(x)))/((1 + (x)^(2))^(s/ 2)*(exp(2*Pi*x)- 1)), x = 0..infinity)	Zeta[s]=Divide[1,2]+Divide[1,s - 1]+ 2*Integrate[Divide[Sin[s*ArcTan[x]],(1 + (x)^(2))^(s/ 2)*(Exp[2*Pi*x]- 1)], {x, 0, Infinity}]	Failure	Successful	Skip	-
25.5.E12	${\displaystyle{\displaystyle\zeta\left(s\right)=\frac{2^{s-1}}{s-1}-2^{s}\int_{0}^{\infty}\frac{\sin\left(s\operatorname{arctan}x\right)}{(1+x^{2})^{s/2}(e^{\pi x}+1)}\mathrm{d}x}}$	Zeta(s)=((2)^(s - 1))/(s - 1)- (2)^(s)* int((sin(s*arctan(x)))/((1 + (x)^(2))^(s/ 2)*(exp(Pi*x)+ 1)), x = 0..infinity)	Zeta[s]=Divide[(2)^(s - 1),s - 1]- (2)^(s)* Integrate[Divide[Sin[s*ArcTan[x]],(1 + (x)^(2))^(s/ 2)*(Exp[Pi*x]+ 1)], {x, 0, Infinity}]	Failure	Successful	Skip	-
25.5.E13	${\displaystyle{\displaystyle\zeta\left(s\right)=\frac{\pi^{s/2}}{s(s-1)\Gamma\left(\frac{1}{2}s\right)}+\frac{\pi^{s/2}}{\Gamma\left(\frac{1}{2}s\right)}\*\int_{1}^{\infty}\left(x^{s/2}+x^{(1-s)/2}\right)\frac{\omega(x)}{x}\mathrm{d}x}}$	Zeta(s)=((Pi)^(s/ 2))/(s*(s - 1)* GAMMA((1)/(2)*s))+((Pi)^(s/ 2))/(GAMMA((1)/(2)*s))* int(((x)^(s/ 2)+ (x)^((1 - s)/ 2))*(omega*(x))/(x), x = 1..infinity)	Zeta[s]=Divide[(Pi)^(s/ 2),s*(s - 1)* Gamma[Divide[1,2]*s]]+Divide[(Pi)^(s/ 2),Gamma[Divide[1,2]*s]]* Integrate[((x)^(s/ 2)+ (x)^((1 - s)/ 2))*Divide[\[Omega]*(x),x], {x, 1, Infinity}]	Failure	Failure	Skip	Skip
25.5.E14	${\displaystyle{\displaystyle\omega(x)=\sum_{n=1}^{\infty}e^{-n^{2}\pi x}}}$	omega*(x)= sum(exp(- (n)^(2)* Pi*x), n = 1..infinity)	\[Omega]*(x)= Sum[Exp[- (n)^(2)* Pi*x], {n, 1, Infinity}]	Failure	Failure	Skip	Fail Complex[1.370996156766441, 1.4142135623730951] <- {Rule[x, 1], Rule[ω, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]} Complex[2.826559682002321, 2.8284271247461903] <- {Rule[x, 2], Rule[ω, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]} Complex[4.242559987601715, 4.242640687119286] <- {Rule[x, 3], Rule[ω, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]} Complex[1.370996156766441, -1.4142135623730951] <- {Rule[x, 1], Rule[ω, Times[Complex[1, -1], Power[2, Rational[1, 2]]]]} ... skip entries to safe data
25.5.E14	${\displaystyle{\displaystyle\sum_{n=1}^{\infty}e^{-n^{2}\pi x}=\frac{1}{2}\left(\theta_{3}\left(0\middle|ix\right)-1\right)}}$	sum(exp(- (n)^(2)* Pi*x), n = 1..infinity)=(1)/(2)*(JacobiTheta3(0,exp(I*Pi*I*x))- 1)	Sum[Exp[- (n)^(2)* Pi*x], {n, 1, Infinity}]=Divide[1,2]*(EllipticTheta[3, 0, I*x]- 1)	Failure	Failure	Skip	Successful
25.5.E15	${\displaystyle{\displaystyle\zeta\left(s\right)=\frac{1}{s-1}+\frac{\sin\left(\pi s\right)}{\pi}\*\int_{0}^{\infty}(\ln\left(1+x\right)-\psi\left(1+x\right))x^{-s}\mathrm{d}x}}$	Zeta(s)=(1)/(s - 1)+(sin(Pi*s))/(Pi)* int((ln(1 + x)- Psi(1 + x))* (x)^(- s), x = 0..infinity)	Zeta[s]=Divide[1,s - 1]+Divide[Sin[Pi*s],Pi]* Integrate[(Log[1 + x]- PolyGamma[1 + x])* (x)^(- s), {x, 0, Infinity}]	Failure	Failure	Skip	Error
25.5.E16	${\displaystyle{\displaystyle\zeta\left(s\right)=\frac{1}{s-1}+\frac{\sin\left(\pi s\right)}{\pi(s-1)}\*\int_{0}^{\infty}\left(\frac{1}{1+x}-\psi'\left(1+x\right)\right)x^{1-s}\mathrm{d}x}}$	Zeta(s)=(1)/(s - 1)+(sin(Pi*s))/(Pi*(s - 1))* int(((1)/(1 + x)- subs( temp=1 + x, diff( Psi(temp), temp$(1) ) ))* (x)^(1 - s), x = 0..infinity)	Zeta[s]=Divide[1,s - 1]+Divide[Sin[Pi*s],Pi*(s - 1)]* Integrate[(Divide[1,1 + x]- (D[PolyGamma[temp], {temp, 1}]/.temp-> 1 + x))* (x)^(1 - s), {x, 0, Infinity}]	Failure	Failure	Skip	-
25.5.E17	${\displaystyle{\displaystyle\zeta\left(1+s\right)=\frac{\sin\left(\pi s\right)}{\pi}\int_{0}^{\infty}\left(\gamma+\psi\left(1+x\right)\right)x^{-s-1}\mathrm{d}x}}$	Zeta(1 + s)=(sin(Pi*s))/(Pi)*int((gamma + Psi(1 + x))* (x)^(- s - 1), x = 0..infinity)	Zeta[1 + s]=Divide[Sin[Pi*s],Pi]*Integrate[(EulerGamma + PolyGamma[1 + x])* (x)^(- s - 1), {x, 0, Infinity}]	Failure	Failure	Skip	Error
25.5.E18	${\displaystyle{\displaystyle\zeta\left(1+s\right)=\frac{\sin\left(\pi s\right)}{\pi s}\int_{0}^{\infty}\psi'\left(1+x\right)x^{-s}\mathrm{d}x}}$	Zeta(1 + s)=(sin(Pi*s))/(Pi*s)*int(subs( temp=1 + x, diff( Psi(temp), temp$(1) ) )*(x)^(- s), x = 0..infinity)	Zeta[1 + s]=Divide[Sin[Pi*s],Pi*s]*Integrate[(D[PolyGamma[temp], {temp, 1}]/.temp-> 1 + x)*(x)^(- s), {x, 0, Infinity}]	Failure	Failure	Skip	Skip
25.5.E19	${\displaystyle{\displaystyle\zeta\left(m+s\right)=(-1)^{m-1}\frac{\Gamma\left(s\right)\sin\left(\pi s\right)}{\pi\Gamma\left(m+s\right)}\*\int_{0}^{\infty}{\psi^{(m)}}\left(1+x\right)x^{-s}\mathrm{d}x}}$	Zeta(m + s)=(- 1)^(m - 1)*(GAMMA(s)*sin(Pi*s))/(Pi*GAMMA(m + s))* int(subs( temp=1 + x, diff( Psi(temp), temp$(m) ) )*(x)^(- s), x = 0..infinity)	Zeta[m + s]=(- 1)^(m - 1)*Divide[Gamma[s]*Sin[Pi*s],Pi*Gamma[m + s]]* Integrate[(D[PolyGamma[temp], {temp, m}]/.temp-> 1 + x)*(x)^(- s), {x, 0, Infinity}]	Failure	Failure	Skip	Error
25.5.E20	${\displaystyle{\displaystyle\zeta\left(s\right)=\frac{\Gamma\left(1-s\right)}{2\pi i}\int_{-\infty}^{(0+)}\frac{z^{s-1}}{e^{-z}-1}\mathrm{d}z}}$	Zeta(s)=(GAMMA(1 - s))/(2*Pi*I)*int(((z)^(s - 1))/(exp(- z)- 1), z = - infinity..(0 +))	Zeta[s]=Divide[Gamma[1 - s],2*Pi*I]*Integrate[Divide[(z)^(s - 1),Exp[- z]- 1], {z, - Infinity, (0 +)}]	Error	Failure	-	Error
25.5.E21	${\displaystyle{\displaystyle\zeta\left(s\right)=\frac{\Gamma\left(1-s\right)}{2\pi i(1-2^{1-s})}\*\int_{-\infty}^{(0+)}\frac{z^{s-1}}{e^{-z}+1}\mathrm{d}z}}$	Zeta(s)=(GAMMA(1 - s))/(2*Pi*I*(1 - (2)^(1 - s)))* int(((z)^(s - 1))/(exp(- z)+ 1), z = - infinity..(0 +))	Zeta[s]=Divide[Gamma[1 - s],2*Pi*I*(1 - (2)^(1 - s))]* Integrate[Divide[(z)^(s - 1),Exp[- z]+ 1], {z, - Infinity, (0 +)}]	Error	Failure	-	Error
25.6#Ex1	${\displaystyle{\displaystyle\zeta\left(0\right)=-\frac{1}{2}}}$	Zeta(0)= -(1)/(2)	Zeta[0]= -Divide[1,2]	Successful	Successful	-	-
25.6#Ex2	${\displaystyle{\displaystyle\zeta\left(2\right)=\frac{\pi^{2}}{6}}}$	Zeta(2)=((Pi)^(2))/(6)	Zeta[2]=Divide[(Pi)^(2),6]	Successful	Successful	-	-
25.6#Ex3	${\displaystyle{\displaystyle\zeta\left(4\right)=\frac{\pi^{4}}{90}}}$	Zeta(4)=((Pi)^(4))/(90)	Zeta[4]=Divide[(Pi)^(4),90]	Successful	Successful	-	-
25.6#Ex4	${\displaystyle{\displaystyle\zeta\left(6\right)=\frac{\pi^{6}}{945}}}$	Zeta(6)=((Pi)^(6))/(945)	Zeta[6]=Divide[(Pi)^(6),945]	Successful	Successful	-	-
25.6.E2	${\displaystyle{\displaystyle\zeta\left(2n\right)=\frac{(2\pi)^{2n}}{2(2n)!}\left|B_{2n}\right|}}$	Zeta(2*n)=((2*Pi)^(2*n))/(2*factorial(2*n))*abs(bernoulli(2*n))	Zeta[2*n]=Divide[(2*Pi)^(2*n),2*(2*n)!]*Abs[BernoulliB[2*n]]	Failure	Failure	Successful	Successful
25.6.E3	${\displaystyle{\displaystyle\zeta\left(-n\right)=-\frac{B_{n+1}}{n+1}}}$	Zeta(- n)= -(bernoulli(n + 1))/(n + 1)	Zeta[- n]= -Divide[BernoulliB[n + 1],n + 1]	Failure	Failure	Successful	Successful
25.6.E4	${\displaystyle{\displaystyle\zeta\left(-2n\right)=0}}$	Zeta(- 2*n)= 0	Zeta[- 2*n]= 0	Failure	Failure	Successful	Successful
25.6.E6	${\displaystyle{\displaystyle\zeta\left(2k+1\right)=\frac{(-1)^{k+1}(2\pi)^{2k+1}}{2(2k+1)!}\int_{0}^{1}B_{2k+1}\left(t\right)\cot\left(\pi t\right)\mathrm{d}t}}$	Zeta(2*k + 1)=((- 1)^(k + 1)*(2*Pi)^(2*k + 1))/(2*factorial(2*k + 1))*int(bernoulli(2*k + 1, t)*cot(Pi*t), t = 0..1)	Zeta[2*k + 1]=Divide[(- 1)^(k + 1)*(2*Pi)^(2*k + 1),2*(2*k + 1)!]*Integrate[BernoulliB[2*k + 1, t]*Cot[Pi*t], {t, 0, 1}]	Failure	Failure	Skip	Successful
25.6.E7	${\displaystyle{\displaystyle\zeta\left(2\right)=\int_{0}^{1}\int_{0}^{1}\frac{1}{1-xy}\mathrm{d}x\mathrm{d}y}}$	Zeta(2)= int(int((1)/(1 - x*y), x = 0..1), y = 0..1)	Zeta[2]= Integrate[Integrate[Divide[1,1 - x*y], {x, 0, 1}], {y, 0, 1}]	Successful	Successful	-	-
25.6.E8	${\displaystyle{\displaystyle\zeta\left(2\right)=3\sum_{k=1}^{\infty}\frac{1}{k^{2}\genfrac{(}{)}{0.0pt}{}{2k}{k}}}}$	Zeta(2)= 3*sum((1)/((k)^(2)*binomial(2*k,k)), k = 1..infinity)	Zeta[2]= 3*Sum[Divide[1,(k)^(2)*Binomial[2*k,k]], {k, 1, Infinity}]	Successful	Successful	-	-
25.6.E9	${\displaystyle{\displaystyle\zeta\left(3\right)=\frac{5}{2}\sum_{k=1}^{\infty}\frac{(-1)^{k-1}}{k^{3}\genfrac{(}{)}{0.0pt}{}{2k}{k}}}}$	Zeta(3)=(5)/(2)*sum(((- 1)^(k - 1))/((k)^(3)*binomial(2*k,k)), k = 1..infinity)	Zeta[3]=Divide[5,2]*Sum[Divide[(- 1)^(k - 1),(k)^(3)*Binomial[2*k,k]], {k, 1, Infinity}]	Failure	Successful	Skip	-
25.6.E10	${\displaystyle{\displaystyle\zeta\left(4\right)=\frac{36}{17}\sum_{k=1}^{\infty}\frac{1}{k^{4}\genfrac{(}{)}{0.0pt}{}{2k}{k}}}}$	Zeta(4)=(36)/(17)*sum((1)/((k)^(4)*binomial(2*k,k)), k = 1..infinity)	Zeta[4]=Divide[36,17]*Sum[Divide[1,(k)^(4)*Binomial[2*k,k]], {k, 1, Infinity}]	Failure	Successful	Skip	-
25.6.E11	${\displaystyle{\displaystyle\zeta'\left(0\right)=-\tfrac{1}{2}\ln\left(2\pi\right)}}$	subs( temp=0, diff( Zeta(temp), temp$(1) ) )= -(1)/(2)*ln(2*Pi)	(D[Zeta[temp], {temp, 1}]/.temp-> 0)= -Divide[1,2]*Log[2*Pi]	Successful	Successful	-	-
25.6.E12	${\displaystyle{\displaystyle\zeta''\left(0\right)=-\tfrac{1}{2}(\ln\left(2\pi\right))^{2}+\tfrac{1}{2}{\gamma^{2}}-\tfrac{1}{24}\pi^{2}+\gamma_{1}}}$	subs( temp=0, diff( Zeta(temp), temp$(2) ) )= -(1)/(2)*(ln(2*Pi))^(2)+(1)/(2)*(gamma)^(2)-(1)/(24)*(Pi)^(2)+ gamma[1]	(D[Zeta[temp], {temp, 2}]/.temp-> 0)= -Divide[1,2]*(Log[2*Pi])^(2)+Divide[1,2]*(EulerGamma)^(2)-Divide[1,24]*(Pi)^(2)+ Subscript[\[Gamma], 1]	Failure	Failure	Fail -1.487029407-1.414213562*I <- {gamma[1] = 2^(1/2)+I*2^(1/2)} -1.487029407+1.414213562*I <- {gamma[1] = 2^(1/2)-I*2^(1/2)} 1.341397717+1.414213562*I <- {gamma[1] = -2^(1/2)-I*2^(1/2)} 1.341397717-1.414213562*I <- {gamma[1] = -2^(1/2)+I*2^(1/2)}	Fail Complex[-1.487029407856772, -1.4142135623730951] <- {Rule[Subscript[γ, 1], Times[Complex[1, 1], Power[2, Rational[1, 2]]]]} Complex[-1.487029407856772, 1.4142135623730951] <- {Rule[Subscript[γ, 1], Times[Complex[1, -1], Power[2, Rational[1, 2]]]]} Complex[1.3413977168894184, 1.4142135623730951] <- {Rule[Subscript[γ, 1], Times[Complex[-1, -1], Power[2, Rational[1, 2]]]]} Complex[1.3413977168894184, -1.4142135623730951] <- {Rule[Subscript[γ, 1], Times[Complex[-1, 1], Power[2, Rational[1, 2]]]]}
25.6.E13	${\displaystyle{\displaystyle(-1)^{k}{\zeta^{(k)}}\left(-2n\right)=\frac{2(-1)^{n}}{(2\pi)^{2n+1}}\sum_{m=0}^{k}\sum_{r=0}^{m}\genfrac{(}{)}{0.0pt}{}{k}{m}\genfrac{(}{)}{0.0pt}{}{m}{r}\Im(c^{k-m})\*{\Gamma^{(r)}}\left(2n+1\right){\zeta^{(m-r)}}\left(2n+1\right)}}$	(- 1)^(k)* subs( temp=- 2*n, diff( Zeta(temp), temp$(k) ) )=(2*(- 1)^(n))/((2*Pi)^(2*n + 1))*sum(sum(binomial(k,m)*binomial(m,r)*Im((c)^(k - m))* subs( temp=2*n + 1, diff( GAMMA(temp), temp$(r) ) )*subs( temp=2*n + 1, diff( Zeta(temp), temp$(m - r) ) ), r = 0..m), m = 0..k)	(- 1)^(k)* (D[Zeta[temp], {temp, k}]/.temp-> - 2*n)=Divide[2*(- 1)^(n),(2*Pi)^(2*n + 1)]*Sum[Sum[Binomial[k,m]*Binomial[m,r]*Im[(c)^(k - m)]* (D[Gamma[temp], {temp, r}]/.temp-> 2*n + 1)*(D[Zeta[temp], {temp, m - r}]/.temp-> 2*n + 1), {r, 0, m}], {m, 0, k}]	Failure	Failure	Skip	Skip
25.6.E14	${\displaystyle{\displaystyle(-1)^{k}{\zeta^{(k)}}\left(1-2n\right)=\frac{2(-1)^{n}}{(2\pi)^{2n}}\sum_{m=0}^{k}\sum_{r=0}^{m}\genfrac{(}{)}{0.0pt}{}{k}{m}\genfrac{(}{)}{0.0pt}{}{m}{r}\Re(c^{k-m})\*{\Gamma^{(r)}}\left(2n\right){\zeta^{(m-r)}}\left(2n\right)}}$	(- 1)^(k)* subs( temp=1 - 2*n, diff( Zeta(temp), temp$(k) ) )=(2*(- 1)^(n))/((2*Pi)^(2*n))*sum(sum(binomial(k,m)*binomial(m,r)*Re((c)^(k - m))* subs( temp=2*n, diff( GAMMA(temp), temp$(r) ) )*subs( temp=2*n, diff( Zeta(temp), temp$(m - r) ) ), r = 0..m), m = 0..k)	(- 1)^(k)* (D[Zeta[temp], {temp, k}]/.temp-> 1 - 2*n)=Divide[2*(- 1)^(n),(2*Pi)^(2*n)]*Sum[Sum[Binomial[k,m]*Binomial[m,r]*Re[(c)^(k - m)]* (D[Gamma[temp], {temp, r}]/.temp-> 2*n)*(D[Zeta[temp], {temp, m - r}]/.temp-> 2*n), {r, 0, m}], {m, 0, k}]	Failure	Failure	Skip	Skip
25.6.E15	${\displaystyle{\displaystyle\zeta'\left(2n\right)=\frac{(-1)^{n+1}(2\pi)^{2n}}{2(2n)!}\left(2n\zeta'\left(1-2n\right)-(\psi\left(2n\right)-\ln\left(2\pi\right))B_{2n}\right)}}$	subs( temp=2*n, diff( Zeta(temp), temp$(1) ) )=((- 1)^(n + 1)*(2*Pi)^(2*n))/(2*factorial(2*n))*(2*n*subs( temp=1 - 2*n, diff( Zeta(temp), temp$(1) ) )-(Psi(2*n)- ln(2*Pi))*bernoulli(2*n))	(D[Zeta[temp], {temp, 1}]/.temp-> 2*n)=Divide[(- 1)^(n + 1)*(2*Pi)^(2*n),2*(2*n)!]*(2*n*(D[Zeta[temp], {temp, 1}]/.temp-> 1 - 2*n)-(PolyGamma[2*n]- Log[2*Pi])*BernoulliB[2*n])	Failure	Failure	Successful	Successful
25.6.E16	${\displaystyle{\displaystyle\left(n+\tfrac{1}{2}\right)\zeta\left(2n\right)=\sum_{k=1}^{n-1}\zeta\left(2k\right)\zeta\left(2n-2k\right)}}$	(n +(1)/(2))* Zeta(2*n)= sum(Zeta(2*k)*Zeta(2*n - 2*k), k = 1..n - 1)	(n +Divide[1,2])* Zeta[2*n]= Sum[Zeta[2*k]*Zeta[2*n - 2*k], {k, 1, n - 1}]	Failure	Failure	Skip	Successful
25.6.E17	${\displaystyle{\displaystyle\left(n+\tfrac{3}{4}\right)\zeta\left(4n+2\right)=\sum_{k=1}^{n}\zeta\left(2k\right)\zeta\left(4n+2-2k\right)}}$	(n +(3)/(4))* Zeta(4*n + 2)= sum(Zeta(2*k)*Zeta(4*n + 2 - 2*k), k = 1..n)	(n +Divide[3,4])* Zeta[4*n + 2]= Sum[Zeta[2*k]*Zeta[4*n + 2 - 2*k], {k, 1, n}]	Failure	Failure	Skip	Successful
25.6.E20	${\displaystyle{\displaystyle\tfrac{1}{2}(2^{2n}-1)\zeta\left(2n\right)=\sum_{k=1}^{n-1}(2^{2n-2k}-1)\zeta\left(2n-2k\right)\zeta\left(2k\right)}}$	(1)/(2)*((2)^(2*n)- 1)* Zeta(2*n)= sum(((2)^(2*n - 2*k)- 1)* Zeta(2*n - 2*k)*Zeta(2*k), k = 1..n - 1)	Divide[1,2]*((2)^(2*n)- 1)* Zeta[2*n]= Sum[((2)^(2*n - 2*k)- 1)* Zeta[2*n - 2*k]*Zeta[2*k], {k, 1, n - 1}]	Failure	Failure	Skip	Successful
25.8.E1	${\displaystyle{\displaystyle\sum_{k=2}^{\infty}\left(\zeta\left(k\right)-1\right)=1}}$	sum(Zeta(k)- 1, k = 2..infinity)= 1	Sum[Zeta[k]- 1, {k, 2, Infinity}]= 1	Failure	Successful	Skip	-
25.8.E2	${\displaystyle{\displaystyle\sum_{k=0}^{\infty}\frac{\Gamma\left(s+k\right)}{(k+1)!}\left(\zeta\left(s+k\right)-1\right)=\Gamma\left(s-1\right)}}$	sum((GAMMA(s + k))/(factorial(k + 1))*(Zeta(s + k)- 1), k = 0..infinity)= GAMMA(s - 1)	Sum[Divide[Gamma[s + k],(k + 1)!]*(Zeta[s + k]- 1), {k, 0, Infinity}]= Gamma[s - 1]	Failure	Failure	Skip	Error
25.8.E3	${\displaystyle{\displaystyle\sum_{k=0}^{\infty}\frac{{\left(s\right)_{k}}\zeta\left(s+k\right)}{k!2^{s+k}}=(1-2^{-s})\zeta\left(s\right)}}$	sum((pochhammer(s, k)*Zeta(s + k))/(factorial(k)*(2)^(s + k)), k = 0..infinity)=(1 - (2)^(- s))* Zeta(s)	Sum[Divide[Pochhammer[s, k]*Zeta[s + k],(k)!*(2)^(s + k)], {k, 0, Infinity}]=(1 - (2)^(- s))* Zeta[s]	Failure	Failure	Skip	Successful
25.8.E4	${\displaystyle{\displaystyle\sum_{k=1}^{\infty}\frac{(-1)^{k}}{k}(\zeta\left(nk\right)-1)=\ln\left(\prod_{j=0}^{n-1}\Gamma\left(2-e^{(2j+1)\pi i/n}\right)\right)}}$	sum(((- 1)^(k))/(k)*(Zeta(n*k)- 1), k = 1..infinity)= ln(product(GAMMA(2 - exp((2*j + 1)* Pi*I/ n)), j = 0..n - 1))	Sum[Divide[(- 1)^(k),k]*(Zeta[n*k]- 1), {k, 1, Infinity}]= Log[Product[Gamma[2 - Exp[(2*j + 1)* Pi*I/ n]], {j, 0, n - 1}]]	Failure	Failure	Skip	Fail Complex[0.7210663818131499, 0.6288153989756469] <- {Rule[Product[Gamma[Plus[2, Times[-1, Power[E, Times[Complex[0, 1], Plus[1, Times[2, j]], Power[n, -1], Pi]]]]], {j, 0, Plus[-1, n]}], Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[Sum[Times[Power[-1, k], Power[k, -1], Plus[-1, Zeta[Times[k, n]]]], {k, 1, DirectedInfinity[1]}], Times[Complex[1, 1], Power[2, Rational[1, 2]]]]} Complex[0.7210663818131499, -2.199611725770543] <- {Rule[Product[Gamma[Plus[2, Times[-1, Power[E, Times[Complex[0, 1], Plus[1, Times[2, j]], Power[n, -1], Pi]]]]], {j, 0, Plus[-1, n]}], Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[Sum[Times[Power[-1, k], Power[k, -1], Plus[-1, Zeta[Times[k, n]]]], {k, 1, DirectedInfinity[1]}], Times[Complex[1, -1], Power[2, Rational[1, 2]]]]} Complex[-2.1073607429330403, -2.199611725770543] <- {Rule[Product[Gamma[Plus[2, Times[-1, Power[E, Times[Complex[0, 1], Plus[1, Times[2, j]], Power[n, -1], Pi]]]]], {j, 0, Plus[-1, n]}], Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[Sum[Times[Power[-1, k], Power[k, -1], Plus[-1, Zeta[Times[k, n]]]], {k, 1, DirectedInfinity[1]}], Times[Complex[-1, -1], Power[2, Rational[1, 2]]]]} Complex[-2.1073607429330403, 0.6288153989756469] <- {Rule[Product[Gamma[Plus[2, Times[-1, Power[E, Times[Complex[0, 1], Plus[1, Times[2, j]], Power[n, -1], Pi]]]]], {j, 0, Plus[-1, n]}], Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[Sum[Times[Power[-1, k], Power[k, -1], Plus[-1, Zeta[Times[k, n]]]], {k, 1, DirectedInfinity[1]}], Times[Complex[-1, 1], Power[2, Rational[1, 2]]]]} ... skip entries to safe data
25.8.E5	${\displaystyle{\displaystyle\sum_{k=2}^{\infty}\zeta\left(k\right)z^{k}=-\gamma z-z\psi\left(1-z\right)}}$	sum(Zeta(k)*(z)^(k), k = 2..infinity)= - gamma*z - z*Psi(1 - z)	Sum[Zeta[k]*(z)^(k), {k, 2, Infinity}]= - EulerGamma*z - z*PolyGamma[1 - z]	Failure	Successful	Skip	-
25.8.E6	${\displaystyle{\displaystyle\sum_{k=0}^{\infty}\zeta\left(2k\right)z^{2k}=-\tfrac{1}{2}\pi z\cot\left(\pi z\right)}}$	sum(Zeta(2*k)*(z)^(2*k), k = 0..infinity)= -(1)/(2)*Pi*z*cot(Pi*z)	Sum[Zeta[2*k]*(z)^(2*k), {k, 0, Infinity}]= -Divide[1,2]*Pi*z*Cot[Pi*z]	Failure	Failure	Skip	Skip
25.8.E7	${\displaystyle{\displaystyle\sum_{k=2}^{\infty}\frac{\zeta\left(k\right)}{k}z^{k}=-\gamma z+\ln\Gamma\left(1-z\right)}}$	sum((Zeta(k))/(k)*(z)^(k), k = 2..infinity)= - gamma*z + ln(GAMMA(1 - z))	Sum[Divide[Zeta[k],k]*(z)^(k), {k, 2, Infinity}]= - EulerGamma*z + Log[Gamma[1 - z]]	Failure	Successful	Skip	-
25.8.E8	${\displaystyle{\displaystyle\sum_{k=1}^{\infty}\frac{\zeta\left(2k\right)}{k}z^{2k}=\ln\left(\frac{\pi z}{\sin\left(\pi z\right)}\right)}}$	sum((Zeta(2*k))/(k)*(z)^(2*k), k = 1..infinity)= ln((Pi*z)/(sin(Pi*z)))	Sum[Divide[Zeta[2*k],k]*(z)^(2*k), {k, 1, Infinity}]= Log[Divide[Pi*z,Sin[Pi*z]]]	Failure	Successful	Skip	-
25.8.E9	${\displaystyle{\displaystyle\sum_{k=1}^{\infty}\frac{\zeta\left(2k\right)}{(2k+1)2^{2k}}=\frac{1}{2}-\frac{1}{2}\ln 2}}$	sum((Zeta(2*k))/((2*k + 1)* (2)^(2*k)), k = 1..infinity)=(1)/(2)-(1)/(2)*ln(2)	Sum[Divide[Zeta[2*k],(2*k + 1)* (2)^(2*k)], {k, 1, Infinity}]=Divide[1,2]-Divide[1,2]*Log[2]	Failure	Successful	Skip	-
25.8.E10	${\displaystyle{\displaystyle\sum_{k=1}^{\infty}\frac{\zeta\left(2k\right)}{(2k+1)(2k+2)2^{2k}}=\frac{1}{4}-\frac{7}{4\pi^{2}}\zeta\left(3\right)}}$	sum((Zeta(2*k))/((2*k + 1)*(2*k + 2)* (2)^(2*k)), k = 1..infinity)=(1)/(4)-(7)/(4*(Pi)^(2))*Zeta(3)	Sum[Divide[Zeta[2*k],(2*k + 1)*(2*k + 2)* (2)^(2*k)], {k, 1, Infinity}]=Divide[1,4]-Divide[7,4*(Pi)^(2)]*Zeta[3]	Failure	Successful	Skip	-
25.9.E2	${\displaystyle{\displaystyle\chi(s)=\pi^{s-\frac{1}{2}}\Gamma\left(\tfrac{1}{2}-\tfrac{1}{2}s\right)/\Gamma\left(\tfrac{1}{2}s\right)}}$	chi*(s)= (Pi)^(s -(1)/(2))* GAMMA((1)/(2)-(1)/(2)*s)/ GAMMA((1)/(2)*s)	\[Chi]*(s)= (Pi)^(s -Divide[1,2])* Gamma[Divide[1,2]-Divide[1,2]*s]/ Gamma[Divide[1,2]*s]	Failure	Failure	Fail .5066144201+7.721862512*I <- {chi = 2^(1/2)+I*2^(1/2), s = 2^(1/2)+I*2^(1/2)} 4.506614418-3.721862514*I <- {chi = 2^(1/2)+I*2^(1/2), s = 2^(1/2)-I*2^(1/2)} -.5006270982e-1-4.069033292*I <- {chi = 2^(1/2)+I*2^(1/2), s = -2^(1/2)-I*2^(1/2)} -4.050062708+.6903329420e-1*I <- {chi = 2^(1/2)+I*2^(1/2), s = -2^(1/2)+I*2^(1/2)} ... skip entries to safe data	Fail Complex[0.5066144187413095, 7.721862514810475] <- {Rule[s, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[χ, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]} Complex[4.506614418741309, 3.721862514810475] <- {Rule[s, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[χ, Times[Complex[1, -1], Power[2, Rational[1, 2]]]]} Complex[0.5066144187413095, -0.2781374851895251] <- {Rule[s, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[χ, Times[Complex[-1, -1], Power[2, Rational[1, 2]]]]} Complex[-3.4933855812586905, 3.721862514810475] <- {Rule[s, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[χ, Times[Complex[-1, 1], Power[2, Rational[1, 2]]]]} ... skip entries to safe data
25.10.E1	${\displaystyle{\displaystyle Z(t)=\exp\left(i\vartheta(t)\right)\zeta\left(\tfrac{1}{2}+it\right)}}$	Z*(t)= exp(I*vartheta*(t))*Zeta((1)/(2)+ I*t)	Z*(t)= Exp[I*\[CurlyTheta]*(t)]*Zeta[Divide[1,2]+ I*t]	Failure	Failure	Fail -.1598353599e-2+4.002319388*I <- {Z = 2^(1/2)+I*2^(1/2), t = 2^(1/2)+I*2^(1/2), vartheta = 2^(1/2)+I*2^(1/2)} .1528788606+3.983270213*I <- {Z = 2^(1/2)+I*2^(1/2), t = 2^(1/2)+I*2^(1/2), vartheta = 2^(1/2)-I*2^(1/2)} -4.764624907+10.91400505*I <- {Z = 2^(1/2)+I*2^(1/2), t = 2^(1/2)+I*2^(1/2), vartheta = -2^(1/2)-I*2^(1/2)} -.3879562929e-1+3.851182221*I <- {Z = 2^(1/2)+I*2^(1/2), t = 2^(1/2)+I*2^(1/2), vartheta = -2^(1/2)+I*2^(1/2)} ... skip entries to safe data	Fail Complex[-0.0015983535965552907, 4.002319390307897] <- {Rule[t, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[Z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[ϑ, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]} Complex[0.15287886062247902, 3.9832702156526483] <- {Rule[t, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[Z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[ϑ, Times[Complex[1, -1], Power[2, Rational[1, 2]]]]} Complex[-4.764624919768366, 10.914005063393518] <- {Rule[t, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[Z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[ϑ, Times[Complex[-1, -1], Power[2, Rational[1, 2]]]]} Complex[-0.03879562949747604, 3.8511822226969143] <- {Rule[t, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[Z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[ϑ, Times[Complex[-1, 1], Power[2, Rational[1, 2]]]]} ... skip entries to safe data
25.10.E3	${\displaystyle{\displaystyle Z(t)=2\sum_{n=1}^{m}\frac{\cos\left(\vartheta(t)-t\ln n\right)}{n^{1/2}}+R(t)}}$	Z*(t)= 2*sum((cos(vartheta*(t)- t*ln(n)))/((n)^(1/ 2)), n = 1..m)+ R*(t)	Z*(t)= 2*Sum[Divide[Cos[\[CurlyTheta]*(t)- t*Log[n]],(n)^(1/ 2)], {n, 1, m}]+ R*(t)	Failure	Failure	Skip	Skip
25.11.E1	${\displaystyle{\displaystyle\zeta\left(s,a\right)=\sum_{n=0}^{\infty}\frac{1}{(n+a)^{s}}}}$	Zeta(0, s, a)= sum((1)/((n + a)^(s)), n = 0..infinity)	HurwitzZeta[s, a]= Sum[Divide[1,(n + a)^(s)], {n, 0, Infinity}]	Failure	Successful	Skip	-
25.11.E2	${\displaystyle{\displaystyle\zeta\left(s,1\right)=\zeta\left(s\right)}}$	Zeta(0, s, 1)= Zeta(s)	HurwitzZeta[s, 1]= Zeta[s]	Successful	Successful	-	-
25.11.E3	${\displaystyle{\displaystyle\zeta\left(s,a\right)=\zeta\left(s,a+1\right)+a^{-s}}}$	Zeta(0, s, a)= Zeta(0, s, a + 1)+ (a)^(- s)	HurwitzZeta[s, a]= HurwitzZeta[s, a + 1]+ (a)^(- s)	Failure	Successful	Successful	-
25.11.E4	${\displaystyle{\displaystyle\zeta\left(s,a\right)=\zeta\left(s,a+m\right)+\sum_{n=0}^{m-1}\frac{1}{(n+a)^{s}}}}$	Zeta(0, s, a)= Zeta(0, s, a + m)+ sum((1)/((n + a)^(s)), n = 0..m - 1)	HurwitzZeta[s, a]= HurwitzZeta[s, a + m]+ Sum[Divide[1,(n + a)^(s)], {n, 0, m - 1}]	Failure	Successful	Skip	-
25.11.E5	${\displaystyle{\displaystyle\zeta\left(s,a\right)=\sum_{n=0}^{N}\frac{1}{(n+a)^{s}}+\frac{(N+a)^{1-s}}{s-1}-s\int_{N}^{\infty}\frac{x-\left\lfloor x\right\rfloor}{(x+a)^{s+1}}\mathrm{d}x}}$	Zeta(0, s, a)= sum((1)/((n + a)^(s)), n = 0..N)+((N + a)^(1 - s))/(s - 1)- s*int((x - floor(x))/((x + a)^(s + 1)), x = N..infinity)	HurwitzZeta[s, a]= Sum[Divide[1,(n + a)^(s)], {n, 0, N}]+Divide[(N + a)^(1 - s),s - 1]- s*Integrate[Divide[x - Floor[x],(x + a)^(s + 1)], {x, N, Infinity}]	Failure	Failure	Skip	Error
25.11.E8	${\displaystyle{\displaystyle\zeta\left(s,\tfrac{1}{2}a\right)=\zeta\left(s,\tfrac{1}{2}a+\tfrac{1}{2}\right)+2^{s}\sum_{n=0}^{\infty}\frac{(-1)^{n}}{(n+a)^{s}}}}$	Zeta(0, s, (1)/(2)*a)= Zeta(0, s, (1)/(2)*a +(1)/(2))+ (2)^(s)* sum(((- 1)^(n))/((n + a)^(s)), n = 0..infinity)	HurwitzZeta[s, Divide[1,2]*a]= HurwitzZeta[s, Divide[1,2]*a +Divide[1,2]]+ (2)^(s)* Sum[Divide[(- 1)^(n),(n + a)^(s)], {n, 0, Infinity}]	Failure	Failure	Skip	Successful
25.11.E9	${\displaystyle{\displaystyle\zeta\left(1-s,a\right)=\frac{2\Gamma\left(s\right)}{(2\pi)^{s}}\*\sum_{n=1}^{\infty}\frac{1}{n^{s}}\cos\left(\tfrac{1}{2}\pi s-2n\pi a\right)}}$	Zeta(0, 1 - s, a)=(2*GAMMA(s))/((2*Pi)^(s))* sum((1)/((n)^(s))*cos((1)/(2)*Pi*s - 2*n*Pi*a), n = 1..infinity)	HurwitzZeta[1 - s, a]=Divide[2*Gamma[s],(2*Pi)^(s)]* Sum[Divide[1,(n)^(s)]*Cos[Divide[1,2]*Pi*s - 2*n*Pi*a], {n, 1, Infinity}]	Failure	Failure	Skip	Error
25.11.E10	${\displaystyle{\displaystyle\zeta\left(s,a\right)=\sum_{n=0}^{\infty}\frac{{\left(s\right)_{n}}}{n!}\zeta\left(n+s\right)(1-a)^{n}}}$	Zeta(0, s, a)= sum((pochhammer(s, n))/(factorial(n))*Zeta(n + s)*(1 - a)^(n), n = 0..infinity)	HurwitzZeta[s, a]= Sum[Divide[Pochhammer[s, n],(n)!]*Zeta[n + s]*(1 - a)^(n), {n, 0, Infinity}]	Failure	Failure	Skip	Error
25.11.E11	${\displaystyle{\displaystyle\zeta\left(s,\tfrac{1}{2}\right)=(2^{s}-1)\zeta\left(s\right)}}$	Zeta(0, s, (1)/(2))=((2)^(s)- 1)* Zeta(s)	HurwitzZeta[s, Divide[1,2]]=((2)^(s)- 1)* Zeta[s]	Successful	Failure	-	Successful
25.11.E12	${\displaystyle{\displaystyle\zeta\left(n+1,a\right)=\frac{(-1)^{n+1}{\psi^{(n)}}\left(a\right)}{n!}}}$	Zeta(0, n + 1, a)=((- 1)^(n + 1)* subs( temp=a, diff( Psi(temp), temp$(n) ) ))/(factorial(n))	HurwitzZeta[n + 1, a]=Divide[(- 1)^(n + 1)* (D[PolyGamma[temp], {temp, n}]/.temp-> a),(n)!]	Failure	Failure	Successful	Successful
25.11.E13	${\displaystyle{\displaystyle\zeta\left(0,a\right)=\tfrac{1}{2}-a}}$	Zeta(0, 0, a)=(1)/(2)- a	HurwitzZeta[0, a]=Divide[1,2]- a	Successful	Successful	-	-
25.11.E14	${\displaystyle{\displaystyle\zeta\left(-n,a\right)=-\frac{B_{n+1}\left(a\right)}{n+1}}}$	Zeta(0, - n, a)= -(bernoulli(n + 1, a))/(n + 1)	HurwitzZeta[- n, a]= -Divide[BernoulliB[n + 1, a],n + 1]	Failure	Failure	Successful	Successful
25.11.E15	${\displaystyle{\displaystyle\zeta\left(s,ka\right)=k^{-s}\*\sum_{n=0}^{k-1}\zeta\left(s,a+\frac{n}{k}\right)}}$	Zeta(0, s, k*a)= (k)^(- s)* sum(Zeta(0, s, a +(n)/(k)), n = 0..k - 1)	HurwitzZeta[s, k*a]= (k)^(- s)* Sum[HurwitzZeta[s, a +Divide[n,k]], {n, 0, k - 1}]	Failure	Failure	Skip	Error
25.11.E16	${\displaystyle{\displaystyle\zeta\left(1-s,\frac{h}{k}\right)=\frac{2\Gamma\left(s\right)}{(2\pi k)^{s}}\*\sum_{r=1}^{k}\cos\left(\frac{\pi s}{2}-\frac{2\pi rh}{k}\right)\zeta\left(s,\frac{r}{k}\right)}}$	Zeta(0, 1 - s, (h)/(k))=(2*GAMMA(s))/((2*Pi*k)^(s))* sum(cos((Pi*s)/(2)-(2*Pi*r*h)/(k))*Zeta(0, s, (r)/(k)), r = 1..k)	HurwitzZeta[1 - s, Divide[h,k]]=Divide[2*Gamma[s],(2*Pi*k)^(s)]* Sum[Cos[Divide[Pi*s,2]-Divide[2*Pi*r*h,k]]*HurwitzZeta[s, Divide[r,k]], {r, 1, k}]	Failure	Failure	Skip	Error
25.11.E17	${\displaystyle{\displaystyle\frac{\partial}{\partial a}\zeta\left(s,a\right)=-s\zeta\left(s+1,a\right)}}$	diff(Zeta(0, s, a), a)= - s*Zeta(0, s + 1, a)	D[HurwitzZeta[s, a], a]= - s*HurwitzZeta[s + 1, a]	Successful	Successful	-	-
25.11.E18	${\displaystyle{\displaystyle\zeta'\left(0,a\right)=\ln\Gamma\left(a\right)-\tfrac{1}{2}\ln\left(2\pi\right)}}$	subs( temp=0, diff( Zeta(0, temp, a), temp$(1) ) )= ln(GAMMA(a))-(1)/(2)*ln(2*Pi)	(D[HurwitzZeta[temp, a], {temp, 1}]/.temp-> 0)= Log[Gamma[a]]-Divide[1,2]*Log[2*Pi]	Failure	Failure	Successful	Successful
25.11.E21	${\displaystyle{\displaystyle\zeta'\left(1-2n,\frac{h}{k}\right)=\frac{(\psi\left(2n\right)-\ln\left(2\pi k\right))B_{2n}\left(h/k\right)}{2n}-\frac{(\psi\left(2n\right)-\ln\left(2\pi\right))B_{2n}}{2nk^{2n}}+\frac{(-1)^{n+1}\pi}{(2\pi k)^{2n}}\sum_{r=1}^{k-1}\sin\left(\frac{2\pi rh}{k}\right){\psi^{(2n-1)}}\left(\frac{r}{k}\right)+\frac{(-1)^{n+1}2\cdot(2n-1)!}{(2\pi k)^{2n}}\sum_{r=1}^{k-1}\cos\left(\frac{2\pi rh}{k}\right)\zeta'\left(2n,\frac{r}{k}\right)+\frac{\zeta'\left(1-2n\right)}{k^{2n}}}}$	subs( temp=1 - 2*n, diff( Zeta(0, temp, (h)/(k)), temp$(1) ) )=((Psi(2*n)- ln(2*Pi*k))* bernoulli(2*n, h/ k))/(2*n)-((Psi(2*n)- ln(2*Pi))* bernoulli(2*n))/(2*n*(k)^(2*n))+((- 1)^(n + 1)* Pi)/((2*Pi*k)^(2*n))*sum(sin((2*Pi*r*h)/(k))*subs( temp=(r)/(k), diff( Psi(temp), temp$(2*n - 1) ) ), r = 1..k - 1)+((- 1)^(n + 1)* 2 *factorial(2*n - 1))/((2*Pi*k)^(2*n))*sum(cos((2*Pi*r*h)/(k))*subs( temp=2*n, diff( Zeta(0, temp, (r)/(k)), temp$(1) ) ), r = 1..k - 1)+(subs( temp=1 - 2*n, diff( Zeta(temp), temp$(1) ) ))/((k)^(2*n))	(D[HurwitzZeta[temp, Divide[h,k]], {temp, 1}]/.temp-> 1 - 2*n)=Divide[(PolyGamma[2*n]- Log[2*Pi*k])* BernoulliB[2*n, h/ k],2*n]-Divide[(PolyGamma[2*n]- Log[2*Pi])* BernoulliB[2*n],2*n*(k)^(2*n)]+Divide[(- 1)^(n + 1)* Pi,(2*Pi*k)^(2*n)]*Sum[Sin[Divide[2*Pi*r*h,k]]*(D[PolyGamma[temp], {temp, 2*n - 1}]/.temp-> Divide[r,k]), {r, 1, k - 1}]+Divide[(- 1)^(n + 1)* 2 *(2*n - 1)!,(2*Pi*k)^(2*n)]*Sum[Cos[Divide[2*Pi*r*h,k]]*(D[HurwitzZeta[temp, Divide[r,k]], {temp, 1}]/.temp-> 2*n), {r, 1, k - 1}]+Divide[D[Zeta[temp], {temp, 1}]/.temp-> 1 - 2*n,(k)^(2*n)]	Failure	Failure	Skip	Error
25.11.E22	${\displaystyle{\displaystyle\zeta'\left(1-2n,\tfrac{1}{2}\right)=-\frac{B_{2n}\ln 2}{n\cdot 4^{n}}-\frac{(2^{2n-1}-1)\zeta'\left(1-2n\right)}{2^{2n-1}}}}$	subs( temp=1 - 2*n, diff( Zeta(0, temp, (1)/(2)), temp$(1) ) )= -(bernoulli(2*n)*ln(2))/(n * (4)^(n))-(((2)^(2*n - 1)- 1)* subs( temp=1 - 2*n, diff( Zeta(temp), temp$(1) ) ))/((2)^(2*n - 1))	(D[HurwitzZeta[temp, Divide[1,2]], {temp, 1}]/.temp-> 1 - 2*n)= -Divide[BernoulliB[2*n]*Log[2],n * (4)^(n)]-Divide[((2)^(2*n - 1)- 1)* (D[Zeta[temp], {temp, 1}]/.temp-> 1 - 2*n),(2)^(2*n - 1)]	Failure	Failure	Successful	Successful
25.11.E23	${\displaystyle{\displaystyle\zeta'\left(1-2n,\tfrac{1}{3}\right)=-\frac{\pi(9^{n}-1)B_{2n}}{8n\sqrt{3}(3^{2n-1}-1)}-\frac{B_{2n}\ln 3}{4n\cdot 3^{2n-1}}-\frac{(-1)^{n}{\psi^{(2n-1)}}\left(\frac{1}{3}\right)}{2\sqrt{3}(6\pi)^{2n-1}}-\frac{\left(3^{2n-1}-1\right)\zeta'\left(1-2n\right)}{2\cdot 3^{2n-1}}}}$	subs( temp=1 - 2*n, diff( Zeta(0, temp, (1)/(3)), temp$(1) ) )= -(Pi*((9)^(n)- 1)* bernoulli(2*n))/(8*n*sqrt(3)*((3)^(2*n - 1)- 1))-(bernoulli(2*n)*ln(3))/(4*n * (3)^(2*n - 1))-((- 1)^(n)* subs( temp=(1)/(3), diff( Psi(temp), temp$(2*n - 1) ) ))/(2*sqrt(3)*(6*Pi)^(2*n - 1))-(((3)^(2*n - 1)- 1)* subs( temp=1 - 2*n, diff( Zeta(temp), temp$(1) ) ))/(2 * (3)^(2*n - 1))	(D[HurwitzZeta[temp, Divide[1,3]], {temp, 1}]/.temp-> 1 - 2*n)= -Divide[Pi*((9)^(n)- 1)* BernoulliB[2*n],8*n*Sqrt[3]*((3)^(2*n - 1)- 1)]-Divide[BernoulliB[2*n]*Log[3],4*n * (3)^(2*n - 1)]-Divide[(- 1)^(n)* (D[PolyGamma[temp], {temp, 2*n - 1}]/.temp-> Divide[1,3]),2*Sqrt[3]*(6*Pi)^(2*n - 1)]-Divide[((3)^(2*n - 1)- 1)* (D[Zeta[temp], {temp, 1}]/.temp-> 1 - 2*n),2 * (3)^(2*n - 1)]	Failure	Failure	Successful	Successful
25.11.E24	${\displaystyle{\displaystyle\sum_{r=1}^{k-1}\zeta'\left(s,\frac{r}{k}\right)=(k^{s}-1)\zeta'\left(s\right)+k^{s}\zeta\left(s\right)\ln k}}$	sum(subs( temp=s, diff( Zeta(0, temp, (r)/(k)), temp$(1) ) ), r = 1..k - 1)=((k)^(s)- 1)* subs( temp=s, diff( Zeta(temp), temp$(1) ) )+ (k)^(s)* Zeta(s)*ln(k)	Sum[D[HurwitzZeta[temp, Divide[r,k]], {temp, 1}]/.temp-> s, {r, 1, k - 1}]=((k)^(s)- 1)* (D[Zeta[temp], {temp, 1}]/.temp-> s)+ (k)^(s)* Zeta[s]*Log[k]	Failure	Failure	Skip	Successful
25.11.E25	${\displaystyle{\displaystyle\zeta\left(s,a\right)=\frac{1}{\Gamma\left(s\right)}\int_{0}^{\infty}\frac{x^{s-1}e^{-ax}}{1-e^{-x}}\mathrm{d}x}}$	Zeta(0, s, a)=(1)/(GAMMA(s))*int(((x)^(s - 1)* exp(- a*x))/(1 - exp(- x)), x = 0..infinity)	HurwitzZeta[s, a]=Divide[1,Gamma[s]]*Integrate[Divide[(x)^(s - 1)* Exp[- a*x],1 - Exp[- x]], {x, 0, Infinity}]	Failure	Failure	Skip	Skip
25.11.E26	${\displaystyle{\displaystyle\zeta\left(s,a\right)=-s\int_{-a}^{\infty}\frac{x-\left\lfloor x\right\rfloor-\frac{1}{2}}{(x+a)^{s+1}}\mathrm{d}x}}$	Zeta(0, s, a)= - s*int((x - floor(x)-(1)/(2))/((x + a)^(s + 1)), x = - a..infinity)	HurwitzZeta[s, a]= - s*Integrate[Divide[x - Floor[x]-Divide[1,2],(x + a)^(s + 1)], {x, - a, Infinity}]	Failure	Failure	Skip	Error
25.11.E27	${\displaystyle{\displaystyle\zeta\left(s,a\right)=\frac{1}{2}a^{-s}+\frac{a^{1-s}}{s-1}+\frac{1}{\Gamma\left(s\right)}\int_{0}^{\infty}\left(\frac{1}{e^{x}-1}-\frac{1}{x}+\frac{1}{2}\right)\frac{x^{s-1}}{e^{ax}}\mathrm{d}x}}$	Zeta(0, s, a)=(1)/(2)*(a)^(- s)+((a)^(1 - s))/(s - 1)+(1)/(GAMMA(s))*int(((1)/(exp(x)- 1)-(1)/(x)+(1)/(2))*((x)^(s - 1))/(exp(a*x)), x = 0..infinity)	HurwitzZeta[s, a]=Divide[1,2]*(a)^(- s)+Divide[(a)^(1 - s),s - 1]+Divide[1,Gamma[s]]*Integrate[(Divide[1,Exp[x]- 1]-Divide[1,x]+Divide[1,2])*Divide[(x)^(s - 1),Exp[a*x]], {x, 0, Infinity}]	Failure	Failure	Skip	Error
25.11.E28	${\displaystyle{\displaystyle\zeta\left(s,a\right)=\frac{1}{2}a^{-s}+\frac{a^{1-s}}{s-1}+\sum_{k=1}^{n}\frac{B_{2k}}{(2k)!}{\left(s\right)_{2k-1}}a^{1-s-2k}+\frac{1}{\Gamma\left(s\right)}\int_{0}^{\infty}\left(\frac{1}{e^{x}-1}-\frac{1}{x}+\frac{1}{2}-\sum_{k=1}^{n}\frac{B_{2k}}{(2k)!}x^{2k-1}\right)x^{s-1}e^{-ax}\mathrm{d}x}}$	Zeta(0, s, a)=(1)/(2)*(a)^(- s)+((a)^(1 - s))/(s - 1)+ sum((bernoulli(2*k))/(factorial(2*k))*pochhammer(s, 2*k - 1)*(a)^(1 - s - 2*k)+(1)/(GAMMA(s))*int(((1)/(exp(x)- 1)-(1)/(x)+(1)/(2)- sum((bernoulli(2*k))/(factorial(2*k))*(x)^(2*k - 1), k = 1..n))* (x)^(s - 1)* exp(- a*x), x = 0..infinity), k = 1..n)	HurwitzZeta[s, a]=Divide[1,2]*(a)^(- s)+Divide[(a)^(1 - s),s - 1]+ Sum[Divide[BernoulliB[2*k],(2*k)!]*Pochhammer[s, 2*k - 1]*(a)^(1 - s - 2*k)+Divide[1,Gamma[s]]*Integrate[(Divide[1,Exp[x]- 1]-Divide[1,x]+Divide[1,2]- Sum[Divide[BernoulliB[2*k],(2*k)!]*(x)^(2*k - 1), {k, 1, n}])* (x)^(s - 1)* Exp[- a*x], {x, 0, Infinity}], {k, 1, n}]	Failure	Failure	Skip	Error
25.11.E29	${\displaystyle{\displaystyle\zeta\left(s,a\right)=\frac{1}{2}a^{-s}+\frac{a^{1-s}}{s-1}+2\int_{0}^{\infty}\frac{\sin\left(s\operatorname{arctan}\left(x/a\right)\right)}{(a^{2}+x^{2})^{s/2}(e^{2\pi x}-1)}\mathrm{d}x}}$	Zeta(0, s, a)=(1)/(2)*(a)^(- s)+((a)^(1 - s))/(s - 1)+ 2*int((sin(s*arctan(x/ a)))/(((a)^(2)+ (x)^(2))^(s/ 2)*(exp(2*Pi*x)- 1)), x = 0..infinity)	HurwitzZeta[s, a]=Divide[1,2]*(a)^(- s)+Divide[(a)^(1 - s),s - 1]+ 2*Integrate[Divide[Sin[s*ArcTan[x/ a]],((a)^(2)+ (x)^(2))^(s/ 2)*(Exp[2*Pi*x]- 1)], {x, 0, Infinity}]	Failure	Failure	Skip	Error
25.11.E30	${\displaystyle{\displaystyle\zeta\left(s,a\right)=\frac{\Gamma\left(1-s\right)}{2\pi i}\int_{-\infty}^{(0+)}\frac{e^{az}z^{s-1}}{1-e^{z}}\mathrm{d}z}}$	Zeta(0, s, a)=(GAMMA(1 - s))/(2*Pi*I)*int((exp(a*z)*(z)^(s - 1))/(1 - exp(z)), z = - infinity..(0 +))	HurwitzZeta[s, a]=Divide[Gamma[1 - s],2*Pi*I]*Integrate[Divide[Exp[a*z]*(z)^(s - 1),1 - Exp[z]], {z, - Infinity, (0 +)}]	Error	Failure	-	Error
25.11.E31	${\displaystyle{\displaystyle\frac{1}{\Gamma\left(s\right)}\int_{0}^{\infty}\frac{x^{s-1}e^{-ax}}{2\cosh x}\mathrm{d}x=4^{-s}\left(\zeta\left(s,\tfrac{1}{4}+\tfrac{1}{4}a\right)-\zeta\left(s,\tfrac{3}{4}+\tfrac{1}{4}a\right)\right)}}$	(1)/(GAMMA(s))*int(((x)^(s - 1)* exp(- a*x))/(2*cosh(x)), x = 0..infinity)= (4)^(- s)*(Zeta(0, s, (1)/(4)+(1)/(4)*a)- Zeta(0, s, (3)/(4)+(1)/(4)*a))	Divide[1,Gamma[s]]*Integrate[Divide[(x)^(s - 1)* Exp[- a*x],2*Cosh[x]], {x, 0, Infinity}]= (4)^(- s)*(HurwitzZeta[s, Divide[1,4]+Divide[1,4]*a]- HurwitzZeta[s, Divide[3,4]+Divide[1,4]*a])	Failure	Failure	Skip	Successful
25.11.E32	${\displaystyle{\displaystyle\int_{0}^{a}x^{n}\psi\left(x\right)\mathrm{d}x=(-1)^{n-1}\zeta'\left(-n\right)+(-1)^{n}h(n)\frac{B_{n+1}}{n+1}-\sum_{k=0}^{n}(-1)^{k}\genfrac{(}{)}{0.0pt}{}{n}{k}h(k)\frac{B_{k+1}(a)}{k+1}a^{n-k}+\sum_{k=0}^{n}(-1)^{k}\genfrac{(}{)}{0.0pt}{}{n}{k}\zeta'\left(-k,a\right)a^{n-k}}}$	int((x)^(n)* Psi(x), x = 0..a)=(- 1)^(n - 1)* subs( temp=- n, diff( Zeta(temp), temp$(1) ) )+(- 1)^(n)* h*(n)*(bernoulli(n + 1))/(n + 1)- sum((- 1)^(k)*binomial(n,k)*h*(k)*(bernoulli(k + 1)*(a))/(k + 1)*(a)^(n - k), k = 0..n)+ sum((- 1)^(k)*binomial(n,k)*subs( temp=- k, diff( Zeta(0, temp, a), temp$(1) ) )*(a)^(n - k), k = 0..n)	Integrate[(x)^(n)* PolyGamma[x], {x, 0, a}]=(- 1)^(n - 1)* (D[Zeta[temp], {temp, 1}]/.temp-> - n)+(- 1)^(n)* h*(n)*Divide[BernoulliB[n + 1],n + 1]- Sum[(- 1)^(k)*Binomial[n,k]*h*(k)*Divide[BernoulliB[k + 1]*(a),k + 1]*(a)^(n - k), {k, 0, n}]+ Sum[(- 1)^(k)*Binomial[n,k]*(D[HurwitzZeta[temp, a], {temp, 1}]/.temp-> - k)*(a)^(n - k), {k, 0, n}]	Failure	Failure	Skip	Error
25.11.E34	${\displaystyle{\displaystyle n\int_{0}^{a}\zeta'\left(1-n,x\right)\mathrm{d}x=\zeta'\left(-n,a\right)-\zeta'\left(-n\right)+\frac{B_{n+1}-B_{n+1}\left(a\right)}{n(n+1)}}}$	n*int(subs( temp=1 - n, diff( Zeta(0, temp, x), temp$(1) ) ), x = 0..a)= subs( temp=- n, diff( Zeta(0, temp, a), temp$(1) ) )- subs( temp=- n, diff( Zeta(temp), temp$(1) ) )+(bernoulli(n + 1)- bernoulli(n + 1, a))/(n*(n + 1))	n*Integrate[D[HurwitzZeta[temp, x], {temp, 1}]/.temp-> 1 - n, {x, 0, a}]= (D[HurwitzZeta[temp, a], {temp, 1}]/.temp-> - n)- (D[Zeta[temp], {temp, 1}]/.temp-> - n)+Divide[BernoulliB[n + 1]- BernoulliB[n + 1, a],n*(n + 1)]	Failure	Failure	Skip	Successful
25.11.E35	${\displaystyle{\displaystyle\sum_{n=0}^{\infty}\frac{(-1)^{n}}{(n+a)^{s}}=\frac{1}{\Gamma\left(s\right)}\int_{0}^{\infty}\frac{x^{s-1}e^{-ax}}{1+e^{-x}}\mathrm{d}x}}$	sum(((- 1)^(n))/((n + a)^(s)), n = 0..infinity)=(1)/(GAMMA(s))*int(((x)^(s - 1)* exp(- a*x))/(1 + exp(- x)), x = 0..infinity)	Sum[Divide[(- 1)^(n),(n + a)^(s)], {n, 0, Infinity}]=Divide[1,Gamma[s]]*Integrate[Divide[(x)^(s - 1)* Exp[- a*x],1 + Exp[- x]], {x, 0, Infinity}]	Error	Failure	-	Skip
25.11.E35	${\displaystyle{\displaystyle\frac{1}{\Gamma\left(s\right)}\int_{0}^{\infty}\frac{x^{s-1}e^{-ax}}{1+e^{-x}}\mathrm{d}x=2^{-s}\left(\zeta\left(s,\tfrac{1}{2}a\right)-\zeta\left(s,\tfrac{1}{2}(1+a)\right)\right)}}$	(1)/(GAMMA(s))*int(((x)^(s - 1)* exp(- a*x))/(1 + exp(- x)), x = 0..infinity)= (2)^(- s)*(Zeta(0, s, (1)/(2)*a)- Zeta(0, s, (1)/(2)*(1 + a)))	Divide[1,Gamma[s]]*Integrate[Divide[(x)^(s - 1)* Exp[- a*x],1 + Exp[- x]], {x, 0, Infinity}]= (2)^(- s)*(HurwitzZeta[s, Divide[1,2]*a]- HurwitzZeta[s, Divide[1,2]*(1 + a)])	Error	Failure	-	Skip
25.11.E36	${\displaystyle{\displaystyle\sum_{n=1}^{\infty}\frac{\chi(n)}{n^{s}}=k^{-s}\sum_{r=1}^{k}\chi(r)\zeta\left(s,\frac{r}{k}\right)}}$	sum((chi*(n))/((n)^(s)), n = 1..infinity)= (k)^(- s)* sum(chi*(r)* Zeta(0, s, (r)/(k)), r = 1..k)	Sum[Divide[\[Chi]*(n),(n)^(s)], {n, 1, Infinity}]= (k)^(- s)* Sum[\[Chi]*(r)* HurwitzZeta[s, Divide[r,k]], {r, 1, k}]	Failure	Failure	Skip	Successful
25.11.E37	${\displaystyle{\displaystyle\sum_{k=1}^{\infty}\frac{(-1)^{k}}{k}\zeta\left(nk,a\right)=-n\ln\Gamma\left(a\right)+\ln\left(\prod_{j=0}^{n-1}\Gamma\left(a-e^{(2j+1)\pi i/n}\right)\right)}}$	sum(((- 1)^(k))/(k)*Zeta(0, n*k, a), k = 1..infinity)= - n*ln(GAMMA(a))+ ln(product(GAMMA(a - exp((2*j + 1)* Pi*I/ n)), j = 0..n - 1))	Sum[Divide[(- 1)^(k),k]*HurwitzZeta[n*k, a], {k, 1, Infinity}]= - n*Log[Gamma[a]]+ Log[Product[Gamma[a - Exp[(2*j + 1)* Pi*I/ n]], {j, 0, n - 1}]]	Failure	Failure	Skip	Skip
25.11.E38	${\displaystyle{\displaystyle\sum_{k=1}^{\infty}\genfrac{(}{)}{0.0pt}{}{n+k}{k}\zeta\left(n+k+1,a\right)z^{k}=\frac{(-1)^{n}}{n!}\left({\psi^{(n)}}\left(a\right)-{\psi^{(n)}}\left(a-z\right)\right)}}$	sum(binomial(n + k,k)*Zeta(0, n + k + 1, a)*(z)^(k), k = 1..infinity)=((- 1)^(n))/(factorial(n))*(subs( temp=a, diff( Psi(temp), temp$(n) ) )- subs( temp=a - z, diff( Psi(temp), temp$(n) ) ))	Sum[Binomial[n + k,k]*HurwitzZeta[n + k + 1, a]*(z)^(k), {k, 1, Infinity}]=Divide[(- 1)^(n),(n)!]*((D[PolyGamma[temp], {temp, n}]/.temp-> a)- (D[PolyGamma[temp], {temp, n}]/.temp-> a - z))	Failure	Failure	Skip	Successful
25.11.E39	${\displaystyle{\displaystyle\sum_{k=2}^{\infty}\frac{k}{2^{k}}\zeta\left(k+1,\tfrac{3}{4}\right)=8G}}$	sum((k)/((2)^(k))*Zeta(0, k + 1, (3)/(4)), k = 2..infinity)= 8*G	Sum[Divide[k,(2)^(k)]*HurwitzZeta[k + 1, Divide[3,4]], {k, 2, Infinity}]= 8*G	Failure	Failure	Skip	Fail Complex[-3.985983745567009, -11.313708498984761] <- {Rule[G, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]} Complex[-3.985983745567009, 11.313708498984761] <- {Rule[G, Times[Complex[1, -1], Power[2, Rational[1, 2]]]]} Complex[18.641433252402514, 11.313708498984761] <- {Rule[G, Times[Complex[-1, -1], Power[2, Rational[1, 2]]]]} Complex[18.641433252402514, -11.313708498984761] <- {Rule[G, Times[Complex[-1, 1], Power[2, Rational[1, 2]]]]}
25.12.E1	${\displaystyle{\displaystyle\mathrm{Li}_{2}\left(z\right)=\sum_{n=1}^{\infty}\frac{z^{n}}{n^{2}}}}$	dilog(z)= sum(((z)^(n))/((n)^(2)), n = 1..infinity)	PolyLog[2, z]= Sum[Divide[(z)^(n),(n)^(2)], {n, 1, Infinity}]	Failure	Successful	Skip	-
25.12.E2	${\displaystyle{\displaystyle\mathrm{Li}_{2}\left(z\right)=-\int_{0}^{z}t^{-1}\ln\left(1-t\right)\mathrm{d}t}}$	dilog(z)= - int((t)^(- 1)* ln(1 - t), t = 0..z)	PolyLog[2, z]= - Integrate[(t)^(- 1)* Log[1 - t], {t, 0, z}]	Failure	Failure	Skip	Error
25.12.E3	${\displaystyle{\displaystyle\mathrm{Li}_{2}\left(z\right)+\mathrm{Li}_{2}\left(\frac{z}{z-1}\right)=-\frac{1}{2}(\ln\left(1-z\right))^{2}}}$	dilog(z)+ dilog((z)/(z - 1))= -(1)/(2)*(ln(1 - z))^(2)	PolyLog[2, z]+ PolyLog[2, Divide[z,z - 1]]= -Divide[1,2]*(Log[1 - z])^(2)	Failure	Failure	Fail 3.289868134-2.177586090*I <- {z = 1/2}	Successful
25.12.E4	${\displaystyle{\displaystyle\mathrm{Li}_{2}\left(z\right)+\mathrm{Li}_{2}\left(\frac{1}{z}\right)=-\frac{1}{6}\pi^{2}-\frac{1}{2}(\ln\left(-z\right))^{2}}}$	dilog(z)+ dilog((1)/(z))= -(1)/(6)*(Pi)^(2)-(1)/(2)*(ln(- z))^(2)	PolyLog[2, z]+ PolyLog[2, Divide[1,z]]= -Divide[1,6]*(Pi)^(2)-Divide[1,2]*(Log[- z])^(2)	Failure	Failure	Fail 6.579736268-4.725198502*I <- {z = -1/2}	Successful
25.12.E5	${\displaystyle{\displaystyle\mathrm{Li}_{2}\left(z^{m}\right)=m\sum_{k=0}^{m-1}\mathrm{Li}_{2}\left(ze^{2\pi ik/m}\right)}}$	dilog((z)^(m))= m*sum(dilog(z*exp(2*Pi*I*k/ m)), k = 0..m - 1)	PolyLog[2, (z)^(m)]= m*Sum[PolyLog[2, z*Exp[2*Pi*I*k/ m]], {k, 0, m - 1}]	Failure	Failure	Skip	Successful
25.12.E6	${\displaystyle{\displaystyle\mathrm{Li}_{2}\left(x\right)+\mathrm{Li}_{2}\left(1-x\right)=\frac{1}{6}\pi^{2}-(\ln x)\ln\left(1-x\right)}}$	dilog(x)+ dilog(1 - x)=(1)/(6)*(Pi)^(2)-(ln(x))* ln(1 - x)	PolyLog[2, x]+ PolyLog[2, 1 - x]=Divide[1,6]*(Pi)^(2)-(Log[x])* Log[1 - x]	Successful	Successful	-	-
25.12.E7	${\displaystyle{\displaystyle\mathrm{Li}_{2}\left(e^{i\theta}\right)=\sum_{n=1}^{\infty}\frac{\cos\left(n\theta\right)}{n^{2}}+i\sum_{n=1}^{\infty}\frac{\sin\left(n\theta\right)}{n^{2}}}}$	dilog(exp(I*theta))= sum((cos(n*theta))/((n)^(2)), n = 1..infinity)+ I*sum((sin(n*theta))/((n)^(2)), n = 1..infinity)	PolyLog[2, Exp[I*\[Theta]]]= Sum[Divide[Cos[n*\[Theta]],(n)^(2)], {n, 1, Infinity}]+ I*Sum[Divide[Sin[n*\[Theta]],(n)^(2)], {n, 1, Infinity}]	Failure	Successful	Skip	-
25.12.E8	${\displaystyle{\displaystyle\sum_{n=1}^{\infty}\frac{\cos\left(n\theta\right)}{n^{2}}=\frac{\pi^{2}}{6}-\frac{\pi\theta}{2}+\frac{\theta^{2}}{4}}}$	sum((cos(n*theta))/((n)^(2)), n = 1..infinity)=((Pi)^(2))/(6)-(Pi*theta)/(2)+((theta)^(2))/(4)	Sum[Divide[Cos[n*\[Theta]],(n)^(2)], {n, 1, Infinity}]=Divide[(Pi)^(2),6]-Divide[Pi*\[Theta],2]+Divide[(\[Theta])^(2),4]	Failure	Failure	Skip	Fail Complex[-4.442882938158366, -4.442882938158366] <- {Rule[θ, Times[Complex[-1, -1], Power[2, Rational[1, 2]]]]} Complex[-4.442882938158366, 4.442882938158366] <- {Rule[θ, Times[Complex[-1, 1], Power[2, Rational[1, 2]]]]}
25.12.E9	${\displaystyle{\displaystyle\sum_{n=1}^{\infty}\frac{\sin\left(n\theta\right)}{n^{2}}=-\int_{0}^{\theta}\ln\left(2\sin\left(\tfrac{1}{2}x\right)\right)\mathrm{d}x}}$	sum((sin(n*theta))/((n)^(2)), n = 1..infinity)= - int(ln(2*sin((1)/(2)*x)), x = 0..theta)	Sum[Divide[Sin[n*\[Theta]],(n)^(2)], {n, 1, Infinity}]= - Integrate[Log[2*Sin[Divide[1,2]*x]], {x, 0, \[Theta]}]	Failure	Failure	Skip	Error
25.12.E10	${\displaystyle{\displaystyle\mathrm{Li}_{s}\left(z\right)=\sum_{n=1}^{\infty}\frac{z^{n}}{n^{s}}}}$	polylog(s, z)= sum(((z)^(n))/((n)^(s)), n = 1..infinity)	PolyLog[s, z]= Sum[Divide[(z)^(n),(n)^(s)], {n, 1, Infinity}]	Failure	Successful	Skip	-
25.12.E11	${\displaystyle{\displaystyle\mathrm{Li}_{s}\left(z\right)=\frac{z}{\Gamma\left(s\right)}\int_{0}^{\infty}\frac{x^{s-1}}{e^{x}-z}\mathrm{d}x}}$	polylog(s, z)=(z)/(GAMMA(s))*int(((x)^(s - 1))/(exp(x)- z), x = 0..infinity)	PolyLog[s, z]=Divide[z,Gamma[s]]*Integrate[Divide[(x)^(s - 1),Exp[x]- z], {x, 0, Infinity}]	Failure	Failure	Skip	Successful
25.12.E12	${\displaystyle{\displaystyle\mathrm{Li}_{s}\left(z\right)=\Gamma\left(1-s\right)\left(\ln\frac{1}{z}\right)^{s-1}+\sum_{n=0}^{\infty}\zeta\left(s-n\right)\frac{(\ln z)^{n}}{n!}}}$	polylog(s, z)= GAMMA(1 - s)*(ln((1)/(z)))^(s - 1)+ sum(Zeta(s - n)*((ln(z))^(n))/(factorial(n)), n = 0..infinity)	PolyLog[s, z]= Gamma[1 - s]*(Log[Divide[1,z]])^(s - 1)+ Sum[Zeta[s - n]*Divide[(Log[z])^(n),(n)!], {n, 0, Infinity}]	Failure	Failure	Skip	Skip
25.12.E13	${\displaystyle{\displaystyle\mathrm{Li}_{s}\left(e^{2\pi ia}\right)+e^{\pi is}\mathrm{Li}_{s}\left(e^{-2\pi ia}\right)=\frac{(2\pi)^{s}e^{\pi is/2}}{\Gamma\left(s\right)}\zeta\left(1-s,a\right)}}$	polylog(s, exp(2*Pi*I*a))+ exp(Pi*I*s)*polylog(s, exp(- 2*Pi*I*a))=((2*Pi)^(s)* exp(Pi*I*s/ 2))/(GAMMA(s))*Zeta(0, 1 - s, a)	PolyLog[s, Exp[2*Pi*I*a]]+ Exp[Pi*I*s]*PolyLog[s, Exp[- 2*Pi*I*a]]=Divide[(2*Pi)^(s)* Exp[Pi*I*s/ 2],Gamma[s]]*HurwitzZeta[1 - s, a]	Failure	Failure	Fail .5737863933-.4240983936*I <- {a = 2^(1/2)+I*2^(1/2), s = 2^(1/2)+I*2^(1/2)} 2281.720763-318.166068*I <- {a = 2^(1/2)+I*2^(1/2), s = 2^(1/2)-I*2^(1/2)} 11.12441999-13.46800186*I <- {a = 2^(1/2)+I*2^(1/2), s = -2^(1/2)-I*2^(1/2)} -.5088647019e-2+.1836981228e-2*I <- {a = 2^(1/2)+I*2^(1/2), s = -2^(1/2)+I*2^(1/2)} ... skip entries to safe data	Fail Complex[0.5737863944300513, -0.4240983930049895] <- {Rule[a, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[s, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]} Complex[2281.720765767148, -318.1660682691354] <- {Rule[a, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[s, Times[Complex[1, -1], Power[2, Rational[1, 2]]]]} Complex[11.124419974397522, -13.468001871634662] <- {Rule[a, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[s, Times[Complex[-1, -1], Power[2, Rational[1, 2]]]]} Complex[-0.00508864702414036, 0.0018369812232921614] <- {Rule[a, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[s, Times[Complex[-1, 1], Power[2, Rational[1, 2]]]]} ... skip entries to safe data
25.12#Ex1	${\displaystyle{\displaystyle F_{s}(x)=-\mathrm{Li}_{s+1}\left(-e^{x}\right)}}$	F[s]*(x)= - polylog(s + 1, - exp(x))	Subscript[F, s]*(x)= - PolyLog[s + 1, - Exp[x]]	Failure	Failure	Fail -.701287556+.9004371571*I <- {s = 2^(1/2)+I*2^(1/2), F[s] = 2^(1/2)+I*2^(1/2), x = 1} -1.490772176+.967006968*I <- {s = 2^(1/2)+I*2^(1/2), F[s] = 2^(1/2)+I*2^(1/2), x = 2} -3.225675211-.894244793*I <- {s = 2^(1/2)+I*2^(1/2), F[s] = 2^(1/2)+I*2^(1/2), x = 3} -.701287556-1.927989967*I <- {s = 2^(1/2)+I*2^(1/2), F[s] = 2^(1/2)-I*2^(1/2), x = 1} ... skip entries to safe data	Successful
25.12#Ex2	${\displaystyle{\displaystyle G_{s}(x)=\mathrm{Li}_{s+1}\left(e^{x}\right)}}$	G[s]*(x)= polylog(s + 1, exp(x))	Subscript[G, s]*(x)= PolyLog[s + 1, Exp[x]]	Failure	Failure	Fail -.592976910+2.518819642*I <- {s = 2^(1/2)+I*2^(1/2), G[s] = 2^(1/2)+I*2^(1/2), x = 1} -.593582812+5.469840344*I <- {s = 2^(1/2)+I*2^(1/2), G[s] = 2^(1/2)+I*2^(1/2), x = 2} -1.275024344+9.341921066*I <- {s = 2^(1/2)+I*2^(1/2), G[s] = 2^(1/2)+I*2^(1/2), x = 3} -.592976910-.309607482*I <- {s = 2^(1/2)+I*2^(1/2), G[s] = 2^(1/2)-I*2^(1/2), x = 1} ... skip entries to safe data	Successful
25.14.E2	${\displaystyle{\displaystyle\zeta\left(s,a\right)=\Phi\left(1,s,a\right)}}$	Zeta(0, s, a)= LerchPhi(1, s, a)	HurwitzZeta[s, a]= LerchPhi[1, s, a]	Successful	Failure	-	Successful
25.14.E3	${\displaystyle{\displaystyle\mathrm{Li}_{s}\left(z\right)=z\Phi\left(z,s,1\right)}}$	polylog(s, z)= z*LerchPhi(z, s, 1)	PolyLog[s, z]= z*LerchPhi[z, s, 1]	Successful	Successful	-	-
25.14.E4	${\displaystyle{\displaystyle\Phi\left(z,s,a\right)=z^{m}\Phi\left(z,s,a+m\right)+\sum_{n=0}^{m-1}\frac{z^{n}}{(a+n)^{s}}}}$	LerchPhi(z, s, a)= (z)^(m)* LerchPhi(z, s, a + m)+ sum(((z)^(n))/((a + n)^(s)), n = 0..m - 1)	LerchPhi[z, s, a]= (z)^(m)* LerchPhi[z, s, a + m]+ Sum[Divide[(z)^(n),(a + n)^(s)], {n, 0, m - 1}]	Failure	Successful	Skip	-
25.14.E5	${\displaystyle{\displaystyle\Phi\left(z,s,a\right)=\frac{1}{\Gamma\left(s\right)}\int_{0}^{\infty}\frac{x^{s-1}e^{-ax}}{1-ze^{-x}}\mathrm{d}x}}$	LerchPhi(z, s, a)=(1)/(GAMMA(s))*int(((x)^(s - 1)* exp(- a*x))/(1 - z*exp(- x)), x = 0..infinity)	LerchPhi[z, s, a]=Divide[1,Gamma[s]]*Integrate[Divide[(x)^(s - 1)* Exp[- a*x],1 - z*Exp[- x]], {x, 0, Infinity}]	Failure	Failure	Skip	Error
25.14.E6	${\displaystyle{\displaystyle\Phi\left(z,s,a\right)=\frac{1}{2}a^{-s}+\int_{0}^{\infty}\frac{z^{x}}{(a+x)^{s}}\mathrm{d}x-2\int_{0}^{\infty}\frac{\sin\left(x\ln z-s\operatorname{arctan}\left(x/a\right)\right)}{(a^{2}+x^{2})^{s/2}(e^{2\pi x}-1)}\mathrm{d}x}}$	LerchPhi(z, s, a)=(1)/(2)*(a)^(- s)+ int(((z)^(x))/((a + x)^(s)), x = 0..infinity)- 2*int((sin(x*ln(z)- s*arctan(x/ a)))/(((a)^(2)+ (x)^(2))^(s/ 2)*(exp(2*Pi*x)- 1)), x = 0..infinity)	LerchPhi[z, s, a]=Divide[1,2]*(a)^(- s)+ Integrate[Divide[(z)^(x),(a + x)^(s)], {x, 0, Infinity}]- 2*Integrate[Divide[Sin[x*Log[z]- s*ArcTan[x/ a]],((a)^(2)+ (x)^(2))^(s/ 2)*(Exp[2*Pi*x]- 1)], {x, 0, Infinity}]	Failure	Failure	Skip	Error
25.16.E10	${\displaystyle{\displaystyle\frac{1}{2}\zeta\left(1-2a\right)=-\frac{B_{2a}}{4a}}}$	(1)/(2)*Zeta(1 - 2*a)= -(bernoulli(2*a))/(4*a)	Divide[1,2]*Zeta[1 - 2*a]= -Divide[BernoulliB[2*a],4*a]	Failure	Failure	Successful	Successful
25.16.E13	${\displaystyle{\displaystyle\sum_{n=1}^{\infty}\left(\frac{h(n)}{n}\right)^{2}=\frac{17}{4}\zeta\left(4\right)}}$	sum(((h*(n))/(n))^(2), n = 1..infinity)=(17)/(4)*Zeta(4)	Sum[(Divide[h*(n),n])^(2), {n, 1, Infinity}]=Divide[17,4]*Zeta[4]	Failure	Failure	Skip	Fail Complex[-3.1856601808992417, 1.4142135623730951] <- {Rule[Sum[Power[h, 2], {n, 1, DirectedInfinity[1]}], Times[Complex[1, 1], Power[2, Rational[1, 2]]]]} Complex[-3.1856601808992417, -1.4142135623730951] <- {Rule[Sum[Power[h, 2], {n, 1, DirectedInfinity[1]}], Times[Complex[1, -1], Power[2, Rational[1, 2]]]]} Complex[-6.014087305645432, -1.4142135623730951] <- {Rule[Sum[Power[h, 2], {n, 1, DirectedInfinity[1]}], Times[Complex[-1, -1], Power[2, Rational[1, 2]]]]} Complex[-6.014087305645432, 1.4142135623730951] <- {Rule[Sum[Power[h, 2], {n, 1, DirectedInfinity[1]}], Times[Complex[-1, 1], Power[2, Rational[1, 2]]]]}
25.16.E15	${\displaystyle{\displaystyle\sum_{r=1}^{\infty}\sum_{k=1}^{r}\frac{1}{r^{2}(r+k)}=\frac{3}{4}\zeta\left(3\right)}}$	sum(sum((1)/((r)^(2)*(r + k)), k = 1..r), r = 1..infinity)=(3)/(4)*Zeta(3)	Sum[Sum[Divide[1,(r)^(2)*(r + k)], {k, 1, r}], {r, 1, Infinity}]=Divide[3,4]*Zeta[3]	Failure	Failure	Skip	Error