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 $\pi$.



Derivative of the Riemann zeta function at $z=-2$
I just came across the article


* Marc-Antoine Coppo, *Generalized Glaisher-Kinkelin constants, Blagouchine’s integrals, and Ramanujan summation*, ([hal-03197403v20](https://hal.univ-cotedazur.fr/hal-03197403v20))


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


