Assuming the grand Riemann hypothesis, we investigate the
distribution of the lowlying zeros of the $L$-functions $L(s,\psi)$,
where $\psi$ is a character of the ideal class group of the imaginary
quadratic field $\mathbb {Q}(\sqrt{-D}) (D\text
{squarefree},D>3,D\equiv 3(\mod 4))$. We prove that, in the
vicinity of the central point $s = 1/2$, the average distribution of
these zeros (for $D\longrightarrow \infty$) is governed by the
symplectic distribution. By averaging over $D$, we go beyond the
natural bound of the support of the Fourier transform of the test
function. This problem is naturally linked with the question of
counting primes $p$ of the form $4p = m\sp 2+Dn\sp 2$, and sieve
techniques are applied.