I was wondering if there is a technique for evaluating the Riemann-zeta function at non-integer points please.
I am aware of the technique from complex analysis that when $s \in \mathbb{N}$, we can construct a function such as
\begin{equation*} f(z) = \begin{cases} \frac{\pi \cot(\pi z)}{z^s} & \quad s \text{ is even} \\[1ex] \frac{\pi \sec(\pi z)}{z^s} & \quad s \text{ is odd} \end{cases} \end{equation*}
And consider the contour integral and Cauchy's residue theorem to deduce the value of $\zeta(s)$.
However, the above argument relies on the fact that the Laurent series of $f$ around $0$ and $\operatorname{res}(f(z), 0)$ are easy to find (namely, we can recall the Laurent expansion for $\cot(\pi z)$, and the residue is precisely the coefficient of the $(s-1)$th term of which).
I am not sure whether this argument still applies, if we let $s$ to be a non-integer parameter, in this case the Laurent series probably won't be that nice.
Thank you in advance!