I am looking for a reference for the moments of $L$-functions evaluated at real parts greater than 1. I have looked and the only reference I can find are concerned with real parts between 0 and 1 (which admittedly is a more interesting question). I feel that this should be known and potentially done a while ago and that I am only finding recent work in my searches.
In particular, I am looking for results that calculate things like
$$\frac{1}{|X|} \sum_{\chi\in X} L(3/2,\chi)$$
where $\chi$ runs over your favourite family of Dirichlet characters, $X$.
I don't think these sorts of things exist in the literature - because they are too easy.
For example, if $X = \{\chi \pmod{q}\}$, then \[\frac{1}{X} \sum_{\chi \in X} L\left(\frac{3}{2},\chi\right) = \sum_{n = 1}^{\infty} \frac{1}{n^{\frac{3}{2}}} \frac{1}{\varphi(q)} \sum_{\chi \pmod{q}} \chi(n).\] By character orthogonality, this is \[\sum_{\substack{n = 1 \\ n \equiv 1 \pmod{q}}}^{\infty} \frac{1}{n^{\frac{3}{2}}},\] which converges absolutely. As $q \to \infty$, this just converges to $1$.