  We prove that more than nine percent of the central values
$L(\frac{1}{2},\chi_p)$ are non-zero, where $p\equiv 1 \pmod{8}$ ranges over
primes and $\chi_p$ is the real primitive Dirichlet character of conductor $p$.
Previously, it was not known whether a positive proportion of these central
values are non-zero. As a by-product, we obtain the order of magnitude of the
second moment of $L(\frac{1}{2},\chi_p)$, and conditionally we obtain the order
of magnitude of the third moment. Assuming the Generalized Riemann Hypothesis,
we show that our lower bound for the second moment is asymptotically sharp.
