  Physicists such as Green, Vanhove, et al show that differential equations
involving automorphic forms govern the behavior of gravitons. One particular
point of interest is solutions to $(\Delta-\lambda)u=E_{\alpha} E_{\beta}$ on
an arithmetic quotient of the exceptional group $E_8$. We establish that the
existence of a solution to $(\Delta-\lambda)u=E_{\alpha}E_{\beta}$ on the
simpler space $SL_2(\mathbb{Z})\backslash SL_2(\mathbb{R})$ for certain values
of $\alpha$ and $\beta$ depends on nontrivial zeros of the Riemann zeta
function $\zeta(s)$. Further, when such a solution exists, we use spectral
theory to solve $(\Delta-\lambda)u=E_{\alpha}E_{\beta}$ on
$SL_2(\mathbb{Z})\backslash SL_2(\mathbb{R})$ and provide proof of the
meromorphic continuation of the solution. The construction of such a solution
uses Arthur truncation, the Maass-Selberg formula, and automorphic Sobolev
spaces.
