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I have a question regarding the derivatives of the Riemann zeta function. It is known that $\zeta'(-1)=\frac{1}{12}-\ln A$, where $A$ is the Glaisher-Kinkelin constant (which is an elegant generalization of $\pi$). This led to the conclusion that $\zeta'(2)=\frac{\pi^2}{6}(\gamma + \ln (2\pi) -12\ln A)$.

I don't understand why similar formulas cannot be found for $\zeta'(-2)$, even though it can be easily established using the Euler-Maclaurin formula to write $\zeta(3)$ in terms of $\pi$ and a constant $A_1$ that generalizes both $A$ and $\pi$.

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  • $\begingroup$ @KConrad : Ok, I'm removing the term 'closed form'. I just want to know if such a formula exists for $\zeta(3)$ or $\zeta'(-2)$" ? $\endgroup$
    – L.L
    Mar 25, 2023 at 20:46
  • $\begingroup$ $\zeta'(-2)=-\zeta(3)/4\pi^2$, and there is no "closed form" for the zeta function at odd positive integers; see for example math.stackexchange.com/q/12815/87355 $\endgroup$
    – Carlo Beenakker
    Mar 25, 2023 at 21:48

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I just came across the article

  • Marc-Antoine Coppo, Generalized Glaisher-Kinkelin constants, Blagouchine’s integrals, and Ramanujan summation, (hal-03197403v20)

(see page 3) that talks about this topic. Indeed, there is a formula.

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