  The Generalized Riemann Hypothesis implies that at least 50% of the central
values $L \left( \frac{1}{2},\chi\right)$ are non-vanishing as $\chi$ ranges
over primitive characters modulo $q$. We show that one may unconditionally go
beyond GRH, in the sense that if one averages over primitive characters modulo
$q$ and averages $q$ over an interval, then at least 50.073% of the central
values are non-vanishing. The proof utilizes the mollification method with a
three-piece mollifier, and relies on estimates for sums of Kloosterman sums due
to Deshouillers and Iwaniec.
