Numerical Computations Concerning Landau-Siegel Zeros

Abstract

We computationally verify that if $L(s,\chi)$ is a quadratic Dirichlet $L$-function modulo $q \leq 10^{10}$ then $L(\sigma,\chi) \neq 0$ for real $\sigma \ge 1-1/(5\log q)$. The number of verified moduli exceeds benchmarks due to Watkins (2004), Platt (2016), and Languasco (2023) by a factor between 66 and 25,000. Our new algorithm draws from zero-free region arguments.