Let $L(s)$ be an L-function, given by its series expansion and admitting an Euler product, say, for $s$ of large enough real part, $$L(s) = \sum_n \frac{a_n}{n^s} = \prod_p \prod_{i=1}^k (1-\alpha_i p^{-s})^{-1}.$$

Assume that $L(s)$ admits a simple pole at $s=x$ with residue $R$, so that it admits a local expansion of the form $$L(s) = \frac{R}{s-x} + H(s),$$

where $H$ is an analytic function around $s=x$. My question is therefore:

Assuming we now explicitly the value of the residue $R$, is there any standard way of computing the value of $H(x)$?

(everything here is quite explicit, but I am still stuck when I try to do something with the cumbersome expression $L(s) - R/(s-x)$...)

NB: If it can help, I am interested in some kind of symmetric square L-functions.