This question quotes from this [article](https://ivv5hpp.uni-muenster.de/u/pschnei/publ/pap/beilin.pdf), 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.



Is there research on the special values of the zeta function outside the integers?
One research paper that might qualify for what you are searching: [On the values of the Riemann zeta-function at rational arguments](https://hal.archives-ouvertes.fr/hal-01109799/document), 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.
> 
> 
> 


