What is $\zeta(\pi)$?
I know that $\operatorname{Re}(\pi)>1$, thus
$$\zeta(\pi)=\sum_{n\geq1}\frac{1}{n^\pi}$$
Yet I have no idea how to even begin evaluating this series. It's probably unrealistic to think that there even is a closed form, but math can be funny that way sometimes.
Edit: for those of you who are confused about the definition of closed form, here's an example: $\zeta(2)=\frac{\pi^2}{6}$. Note that this is purely a ration of two constants, and even though $\pi$ cannot be exactly calculated in a finite number of operations, I am still willing to consider it a closed form. Another example: $$_3F_2\biggr(1,1,\frac{3}{2};\frac{4}{3},\frac{5}{3};\frac{2}{27}\biggl)=\frac{3\pi}{5}-\frac{6\log2}{5}$$
Note that, again, there are constants which require infinite operations to compute, yet if we just see them as constants, there is a finite number of operations in the answer, which is good enough for me. It should be noted that I do not consider a decimal expansion an adequate closed form.
Best Answer
Number theory currently has insufficient tools do deal with this kind of problem. We don't even have a closed form for $\zeta(3)$.