DLMF:25.11.E23 (Q7697)

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DLMF:25.11.E23
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    Statements

    Defining formula
    ζ ′ ⁡ ( 1 - 2 ⁢ n , 1 3 ) = - π ⁢ ( 9 n - 1 ) ⁢ B 2 ⁢ n 8 ⁢ n ⁢ 3 ⁢ ( 3 2 ⁢ n - 1 - 1 ) - B 2 ⁢ n ⁢ ln ⁡ 3 4 ⁢ n ⋅ 3 2 ⁢ n - 1 - ( - 1 ) n ⁢ ψ ( 2 ⁢ n - 1 ) ⁡ ( 1 3 ) 2 ⁢ 3 ⁢ ( 6 ⁢ π ) 2 ⁢ n - 1 - ( 3 2 ⁢ n - 1 - 1 ) ⁢ ζ ′ ⁡ ( 1 - 2 ⁢ n ) 2 ⋅ 3 2 ⁢ n - 1 , diffop Hurwitz-zeta 1 1 2 𝑛 1 3 𝜋 superscript 9 𝑛 1 Bernoulli-number-B 2 𝑛 8 𝑛 3 superscript 3 2 𝑛 1 1 Bernoulli-number-B 2 𝑛 3 ⋅ 4 𝑛 superscript 3 2 𝑛 1 superscript 1 𝑛 digamma 2 𝑛 1 1 3 2 3 superscript 6 𝜋 2 𝑛 1 superscript 3 2 𝑛 1 1 diffop Riemann-zeta 1 1 2 𝑛 ⋅ 2 superscript 3 2 𝑛 1 {\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}},}}
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    DLMF id
    DLMF:25.11.E23
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    constraint
    n = 1 , 2 , 3 , … 𝑛 1 2 3 … {\displaystyle{\displaystyle n=1,2,3,\dots}}
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    Symbols used
    Q11109
    Defining formula
    B n Bernoulli-number-B 𝑛 {\displaystyle{\displaystyle B_{\NVar{n}}}}
    xml-id
    C24.S2.SS1.m1afdec
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    Hurwitz zeta function
    Defining formula
    ζ ⁡ ( s , a ) Hurwitz-zeta 𝑠 𝑎 {\displaystyle{\displaystyle\zeta\left(\NVar{s},\NVar{a}\right)}}
    xml-id
    C25.S11.E1.m2axdec
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