An asymptotic expansion for a Lambert series associated to Siegel cusp forms

In 2000, Hafner and Stopple proved a conjecture of Zagier which states that the constant term of the automorphic function $\mathrm{\mid }\mathrm{\Delta }(x+iy){\mathrm{\mid }}^2$ i.e., the Lambert series ${\sum }_{n=1}^{\mathrm{\infty }}\tau (n{)}^2{e}^{-4\pi ny}$ can be expressed in terms of the non-trivial zeros of the Riemann zeta function. In this article, we study a certain Lambert series associated to Siegel cusp forms and observe a similar phenomenon.