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.

researchalong non-integral arguments. That is, I'm looking for the questions and contexts which motivate people to look for these values, perhaps with no closed-form at all, and not just examples of them. I can accept how my question title can mislead. $\endgroup$ – Matt Cuffaro Mar 6 '19 at 14:23