Is there research on the special values of the zeta function outside the integers?

Question

This question quotes from this article, but I've noticed this pattern in the literature I've read.

"The values or better the leading coefficients at integral arguments of the L-functions of algebraic varieties over number fields seem to be closely related to the global arithmetical geom­etry of these varieties"

That is, the Riemann zeta function takes on special values $$ \zeta(2n)=\mathbb{Q}^\times\times\pi^{2n} \quad \zeta(1-2n)\in\mathbb{Q}^\times $$ for $n\in\mathbb{Z_{>0}}$. The Dedekind zeta function of some number field has a special value at the residue on $s=1$, and etc. etc. up to theory I consider to be cutting edge, e.g. the Beilinson Conjectures and so on. I've even noticed half-integer arguments, but nothing more complicated than that.

Question: Is there any research on the zeta function, Riemann or otherwise, at non-integral values?

EDIT I'm more interested in research that would motivate someone to look for non-integral arguments.

Answer 1

One research paper that might qualify for what you are searching: On the values of the Riemann zeta-function at rational arguments, S. Kanemitsu, Y. Tanigawa and M. Yoshimoto (2001).

We give a closed form evaluation of Ramanujan’s type of the values of the Riemann zeta-function at positive rational arguments in the critical strip.