  We consider the ensemble of $N\times N$ real random symmetric matrices
$H_N^{(R)}$ obtained from the determinant form of the Ihara zeta function
associated to random graphs $\Gamma_N^{(R)}$ of the long-range percolation
radius model with the edge probability determined by a function $\phi(t)$.
  We show that the normalized eigenvalue counting function of $H_N^{( R)}$
weakly converges in average as $N,R\to\infty$, $R=o(N)$ to a unique measure
that depends on the limiting average vertex degree of $\Gamma_N^{(R)}$ given by
$\phi_1 = \int \phi(t) dt$. This measure converges in the limit of infinite
$\phi_1$ to a shift of the Wigner semi-circle distribution. We discuss
relations of these results with the properties of the Ihara zeta function and
weak versions of the graph theory Riemann Hypothesis.
