Zeros of quadratic Dirichlet $L$-functions

We consider Dirichlet $L$-functions $L(s,\chi )$ where $\chi$ is a quadratic character to modulus $q$. We show that for $\chi$ even (resp. odd) $L(\sigma ,\chi )\mathrm{\ne }0$ for all $\sigma >0$ provided that $q\le {10}^9$ (resp. $q\le {10}^{10})$. In particular, this result rules out Siegel–Landau zeros for such $L$-functions. Combined, these results improve on previous work by nearly four orders of magnitude. We also list some applications of this work.