Real zeros of real odd Dirichlet $L$-functions

Abstract

Let χ be a real odd Dirichlet character of modulus d, and let L(s, χ) be the associated Dirichlet L-function. As a consequence of the work of Low and Purdy, it is known that if d ≤ 800000 and d ≠ 115147, 357819, 636184, then L(s, χ) has no positive real zeros. By a simple extension of their ideas and the advantage of thirty years of advances in computational power, we are able to prove that if d ≤ 300 000 000, then L(s, χ) has no positive real zeros.