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

"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.

• related question at math.stackexchange.com/questions/2441075/… Mar 6 '19 at 14:18
• I appreciate this link, especially since it reminds me of the interesting work done on polylogarithms, although I am interested in any research along 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. Mar 6 '19 at 14:23

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.

• Yes, this does look interesting. I'll read it. Mar 6 '19 at 15:08