# DLMF:25.11.E21 (Q7695)

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DLMF:25.11.E21
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## Statements

${\displaystyle{\displaystyle\zeta'\left(1-2n,\frac{h}{k}\right)=\frac{(\psi% \left(2n\right)-\ln\left(2\pi k\right))B_{2n}\left(h/k\right)}{2n}-\frac{(\psi% \left(2n\right)-\ln\left(2\pi\right))B_{2n}}{2nk^{2n}}+\frac{(-1)^{n+1}\pi}{(2% \pi k)^{2n}}\sum_{r=1}^{k-1}\sin\left(\frac{2\pi rh}{k}\right){\psi^{(2n-1)}}% \left(\frac{r}{k}\right)+\frac{(-1)^{n+1}2\cdot(2n-1)!}{(2\pi k)^{2n}}\sum_{r=% 1}^{k-1}\cos\left(\frac{2\pi rh}{k}\right)\zeta'\left(2n,\frac{r}{k}\right)+% \frac{\zeta'\left(1-2n\right)}{k^{2n}},}}$
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DLMF:25.11.E21
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${\displaystyle{\displaystyle B_{\NVar{n}}}}$
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${\displaystyle{\displaystyle B_{\NVar{n}}\left(\NVar{x}\right)}}$
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${\displaystyle{\displaystyle\zeta\left(\NVar{s},\NVar{a}\right)}}$
C25.S11.E1.m2avdec
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${\displaystyle{\displaystyle\zeta\left(\NVar{s}\right)}}$
C25.S2.E1.m2acdec
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${\displaystyle{\displaystyle\pi}}$
C3.S12.E1.m2acdec
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${\displaystyle{\displaystyle\cos\NVar{z}}}$
C4.S14.E2.m2abdec
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${\displaystyle{\displaystyle\psi\left(\NVar{z}\right)}}$
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${\displaystyle{\displaystyle!}}$
introduction.Sx4.p1.t1.r15.m5acdec
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${\displaystyle{\displaystyle\ln\NVar{z}}}$
C4.S2.E2.m2acdec
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${\displaystyle{\displaystyle\sin\NVar{z}}}$
${\displaystyle{\displaystyle n}}$
${\displaystyle{\displaystyle h}}$
${\displaystyle{\displaystyle k}}$