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It is known that the first derivative of the Riemann zeta function $\zeta' (x)$ can be espressed, for $x=2$, as $$\zeta'(2) = -\frac {\pi^2}{6} [ 12 \log(A)- \gamma -\log(2 \pi)]$$ where $A $ is the Glaisher-Kinkelin constant and $\gamma $ is the Euler-Mascheroni constant.

I would be interested to know whether it is possible to express the second derivative $\zeta'' (2)=1.98928... $ in a similar alternative form using other functions and constants.

Anatoly
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    as I understand it, the constant $\ln A$ is - by definition - a regularized version of $\sum_n n^{-s} \ln n$ at $s=-1$, so $\zeta'(-1) = 1/12 - \ln A$ is quite tautologic, and with the functional equation you get your formula for $\zeta'(2)$. – reuns Aug 29 '16 at 22:57
  • and $\Gamma'(1) = -\gamma$ means $\Gamma'(2) = 1-\gamma$, but what about $\Gamma''(1)$ and $\Gamma''(2)$ ? – reuns Aug 29 '16 at 23:07
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    This is equivalent to finding $\zeta''(-1)$, since$$\zeta''(2)=\frac{\pi^2}6\left[\frac{\pi^2}{12}-12\zeta''(-1)-(1-12\ln(A))^2+1+(12\ln(A)-\gamma-\ln(2\pi))^2\right]$$ – Simply Beautiful Art Feb 10 '17 at 02:15

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