We consider the ensemble of real symmetric random matrices $H^{(n,ρ)}$ obtained from the determinant form of the Ihara zeta function of random graphs that have $n$ vertices with the edge probability $ρ/n$. We prove that the normalized eigenvalue counting function of $H^{(n,ρ)}$ weakly converges in average as $n,ρ\to\infty$ and $ρ=o(n^α)$ for any $α>0$ to a shift of the Wigner semi-circle distribution. Our results support a conjecture that the large Erdos-Rényi random graphs satisfy in average the weak graph theory Riemann Hypothesis.