  An explicit Dirichlet series is obtained, which represents an analytic
function of $s$ in the half-plane $\Re s>1/2$ except for having simple poles at
points $s_j$ that correspond to exceptional eigenvalues $\lambda_j$ of the
non-Euclidean Laplacian for Hecke congruence subgroups $\Gamma_0(N)$ by the
relation $\lambda_j=s_j(1-s_j)$ for $j=1,2,..., S$. Coefficients of the
Dirichlet series involve all class numbers $h_d$ of real quadratic number
fields. But, only the terms with $h_d\gg d^{1/2-\epsilon}$ for sufficiently
large discriminants $d$ contribute to the residues $m_j/2$ of the Dirichlet
series at the poles $s_j$, where $m_j$ is the multiplicity of the eigenvalue
$\lambda_j$ for $j=1,2,..., S$. This may indicate (I'm not able to prove yet)
that the multiplicity of exceptional eigenvalues can be arbitrarily large. On
the other hand, by density theorem [3] the multiplicity of exceptional
eigenvalues is bounded above by a constant depending only on $N$.
