  Let $L(s,\chi)$ be the Dirichlet $L$-function associated to a non-principal
primitive character $\chi$ modulo $q$ with $3\le q \le 400\,000$. We prove a
new explicit zero-free region for $L(s,\chi)$: $L(s,\chi)$ does not vanish in
the region $\Re s \ge 1-\frac1{R \log \left(q\max(1,| \Im s|)\right) }$ with
$R=5.60$. This improves a result of McCurley where $9.65$ was shown to be an
admissible value for $R$.
