DLMF:25.11.E19 (Q7693)

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DLMF:25.11.E19
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    Statements

    Defining formula
    ζ ′ ⁡ ( s , a ) = - ln ⁡ a a s ⁢ ( 1 2 + a s - 1 ) - a 1 - s ( s - 1 ) 2 + s ⁢ ( s + 1 ) 2 ⁢ ∫ 0 ∞ ( B ~ 2 ⁡ ( x ) - B 2 ) ⁢ ln ⁡ ( x + a ) ( x + a ) s + 2 ⁢ d x - ( 2 ⁢ s + 1 ) 2 ⁢ ∫ 0 ∞ B ~ 2 ⁡ ( x ) - B 2 ( x + a ) s + 2 ⁢ d x , diffop Hurwitz-zeta 1 𝑠 𝑎 𝑎 superscript 𝑎 𝑠 1 2 𝑎 𝑠 1 superscript 𝑎 1 𝑠 superscript 𝑠 1 2 𝑠 𝑠 1 2 superscript subscript 0 periodic-Bernoulli-polynomial-B 2 𝑥 Bernoulli-number-B 2 𝑥 𝑎 superscript 𝑥 𝑎 𝑠 2 𝑥 2 𝑠 1 2 superscript subscript 0 periodic-Bernoulli-polynomial-B 2 𝑥 Bernoulli-number-B 2 superscript 𝑥 𝑎 𝑠 2 𝑥 {\displaystyle{\displaystyle\zeta'\left(s,a\right)=-\frac{\ln a}{a^{s}}\left(% \frac{1}{2}+\frac{a}{s-1}\right)-\frac{a^{1-s}}{(s-1)^{2}}+\frac{s(s+1)}{2}% \int_{0}^{\infty}\frac{(\widetilde{B}_{2}\left(x\right)-B_{2})\ln\left(x+a% \right)}{(x+a)^{s+2}}\mathrm{d}x-\frac{(2s+1)}{2}\int_{0}^{\infty}\frac{% \widetilde{B}_{2}\left(x\right)-B_{2}}{(x+a)^{s+2}}\mathrm{d}x,}}
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    DLMF id
    DLMF:25.11.E19
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    constraint
    ℜ ⁡ s > - 1 𝑠 1 {\displaystyle{\displaystyle\Re s>-1}}
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    s ≠ 1 𝑠 1 {\displaystyle{\displaystyle s\neq 1}}
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    a > 0 𝑎 0 {\displaystyle{\displaystyle a>0}}
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