  We derive strong and effective lower bounds for the class number h(q) of the
imaginary quadratic field Q(\sqrt{-q}), conditionally subject to the existence
of many small (subnormal) gaps between zeros of the L-function associated with
a character of the class group associated with this field. In particular, we
prove that if the gap between consecutive zeros of the L-function is somewhat
smaller than the average for sufficiently many pairs of zeros on the critical
line, then h >> \sqrt q (log q)^{-A} for some constant A > 0. For the trivial
character, the L-function is the Dedekind zeta-function of the number field and
so contains the Riemann zeta-function as a factor. Thus, as a corollary to our
main result, we prove that this lower bound for h follows from the hypothesis
that there are sufficiently many pairs of adjacent zeros of the Riemann
zeta-function on the critical line whose spacing is slightly smaller than 1/2
of the average spacing.
