  In this paper, we apply the saddle-point method in conjunction with the
theory of the N$\ddot{o}$rlund-Rice integrals to derive a precise asymptotic
formula for the generalized Li coefficients established by Omar and Mazhouda.
Actually, for any function $F$ in the Selberg class $\mathcal{S}$ and under the
Generalized Riemann Hypothesis, we have $$\lambda_{F}(n)=\frac{d_{F}}{2}n\log
n+c_{F}n+O(\sqrt{n}\log n),$$ with
$$c_{F}=\frac{d_{F}}{2}(\gamma-1)+\frac{1}{2}\log(\lambda Q_{F}^{2}),\ \
\lambda=\prod_{j=1}^{r}\lambda_{j}^{2\lambda_{j}},$$ where $\gamma$ is the
Euler constant and the notation is as bellow.
