Let C(q,+) be the set of even, primitive Dirichlet characters (mod q). Using
the mollifier method we show that L(1/2,chi) is not zero for at least half of
the characters chi in C(q,+). Here, L(s,chi) is the Dirichlet L-function
associated to the character chi. This result was previously known to hold for a
third of the chi in C(q,+). In addition, we show that almost all the characters
chi in C(q,+) satisfy L^{(k)}(1/2,chi) is not equal to zero when k and q are
large. Here, L^{(k)}(s,chi) is the k-th derivative of L(s,chi).