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Matt Cuffaro
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Are there special values of the zeta function outside the integers?

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?

Matt Cuffaro
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