Real zeros of real odd Dirichlet $L$-functions

Let $\chi$ be a real odd Dirichlet character of modulus $d$, and let $L(s,\chi)$ be the associated Dirichlet $L$-function. As a consequence of the work of Low and Purdy, it is known that if $d\le 800\,000$ and $d\neq 115\,147$, $357\,819$, $636\,184$, then $L(s,\chi)$ 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 $d\le 300\,000\,000$, then $L(s,\chi)$ has no positive real zeros.