Reflection Formulas

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Reflection Formulas

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Reflection Formulas

${\displaystyle{\displaystyle{\displaystyle\zeta\left(1-s\right)=2(2\pi)^{-s}\cos\left(\tfrac{1}{2}\pi s\right)\Gamma\left(s\right)\RiemannZeta@{s}}}}$

Constraint(s): ${\displaystyle{\displaystyle{\displaystyle s\neq 0,1}}}$

${\displaystyle{\displaystyle{\displaystyle\RiemannZeta@{s}=2(2\pi)^{s-1}\sin\left(\tfrac{1}{2}\pi s\right)\Gamma\left(1-s\right)\RiemannZeta@{1-s}}}}$
${\displaystyle{\displaystyle{\displaystyle\RiemannXi@{s}=\RiemannXi@{1-s}}}}$
${\displaystyle{\displaystyle{\displaystyle\RiemannXi@{s}=\tfrac{1}{2}s(s-1)\Gamma\left(\tfrac{1}{2}s\right){\pi^{-s/2}}\RiemannZeta@{s}}}}$
${\displaystyle{\displaystyle{\displaystyle(-1)^{k}\RiemannZeta^{(k)}@{1-s}=\frac{2}{(2\pi)^{s}}\sum_{m=0}^{k}\sum_{r=0}^{m}\genfrac{(}{)}{0.0pt}{}{k}{m}\genfrac{(}{)}{0.0pt}{}{m}{r}\left(\Re{(c^{k-m})}\cos\left(\tfrac{1}{2}\pi s\right)+\Im{(c^{k-m})}\sin\left(\tfrac{1}{2}\pi s\right)\right){\Gamma^{(r)}}\left(s\right)\RiemannZeta^{(m-r)}@{s}}}}$

Substitution(s): ${\displaystyle{\displaystyle{\displaystyle{\displaystyle c=-\ln\left(2\pi\right)-\tfrac{1}{2}\pi\mathrm{i}}}}}$

Constraint(s): ${\displaystyle{\displaystyle{\displaystyle s\neq 0,1}}}$ &
${\displaystyle{\displaystyle{\displaystyle k=1,2,3,\dots}}}$

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Reflection Formulas

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