Numerical Computations Concerning Landau-Siegel Zeros

Abstract: We computationally verify that if $L(s,χ)$ is a quadratic Dirichlet $L$-function modulo $q \leq 10<sup>{10}$ then $L(σ,χ) \neq 0$ for real $σ\ge 1-1/(5\log q)$. 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.