This is a fairly vague question.

Suppose we have a sequence of positive numbers $(c_n)_n$ and we want to find an asymptotic formula for $S(x) = \sum_{n \leq X} c_n$. In favorable circumstances, standard Tauberian theorems, e.g. the one in Appendix A of Chambert-Loir and Tschinkel's paper "Fonctions zeta des hauteurs des espaces fibres", available at http://www.math.nyu.edu/~tschinke/papers/yuri/01zeta/zeta.pdf, give an asymptotic formula of the form $X^a P(\log X) + O(X^{a - \delta})$, with $P$ a polynomial of degree $b-1$, with $b$ the order of pole at $a$ ($a>0$ is the right most pole). The leading term of the polynomial $P$ is explicit.

Question.Can one explicitly compute the polynomial $P$?

In my particular problem, I have a zeta function $Z(s)= \sum_n c_n.n^{-s}$ which I understand fairly well. One can show that $Z(s) = f(s) + h(s)$ with $f$ written in terms of Eisenstein series with a pole at $a=1$ of order $b=2$, and $h(s)$ harmless. In the Chambert-Loir–Tschinkel theorem $\delta = 1/2$ works, and I have an asymptotic formula of the form $(3/\pi) X \log X + B X + O(X^{1/2})$. Now I'm trying to find the value $B$.