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.