Closed Forms of Certain Zeta constants?

The Riemann Zeta function $\zeta(s)=\sum_{n=1}^\infty \frac{1}{n^s}$ converges for $\operatorname{Re}(s)>1$. I am interested in some particular "irrational " values of it such as:

• $\zeta(\pi)=1.176241738\ldots$,


• $\zeta(e)=1.2690096043\ldots$,


• $\zeta(\sqrt2)=3.020737679\ldots$,


• $\ldots$

Are there closed form representations for these and constants? Are there formulas which consists of these constants?

Answer

There is no reason to suspect that these have a "closed form". There isn't even a known closed form for $\zeta(3)$...