Let $L = \sum a_n n^{-s} $ be the $L$-function of an elliptic curve $E$. I'm trying to get something similar to the von Mangoldt explicit formula that relates $\sum a_n$ with the residues of $\frac{L'}{L}\frac{x^s}{s}$. So far I have
$$\frac{1}{2\pi i}\int_{c-i\infty}^{c+i\infty}-\frac{L'}{L}\frac{x^s}{s}ds = x \text{rank}(E/\mathbb{Q}) - \frac{L'(0)}{L(0)} - \sum_{\rho \text{ nontrivial zero}} \frac{x^\rho}{\rho} - \frac{1}{2}\log(1-\frac{1}{x^2}) $$
The $x \text{rank}(E/\mathbb{Q})$ term is from assuming BSD. Is my math right? Can I evaluate the $\frac{L'(0)}{L(0)}$ term in any way?