  Hafner and Stopple proved a conjecture of Zagier, that the inverse Mellin
transform of the symmetric square $L$-function associated to the Ramanujan tau
function has an asymptotic expansion in terms of the non-trivial zeros of the
Riemann zeta function $\zeta(s)$. Later, Chakraborty, Kanemitsu and the second
author extended this phenomenon for any Hecke eigenform over the full modular
group. In this paper, we study an asymptotic expansion of the Lambert series
\begin{equation*} y^k \sum_{n=1}^\infty \lambda_{f}( n^2 ) \exp (- ny), \quad
\textrm{as}\,\, y \rightarrow 0^{+}, \end{equation*} where $\lambda_f(n)$ is
the $n$th Fourier coefficient of a Hecke eigen form $f(z)$ of weight $k$ over
the full modular group.
