Split Primes and the Exponent of Imaginary Quadratic Class Groups

Assume that no Dirichlet \(L\)-function vanishes in the half-plane \(\Re s>11/12\), as a recent preprint asserts. Then every imaginary quadratic field \(\Q(\sqrt d)\) has a split prime below $(2\log|d|)^{12}$, its class group has exponent at least \(\log(|d|/4)/(12\log(2\log|d|))\), and Euler's sixty-five idoneal numbers are all of them. With \(\Re s>7/8\) the exponent $12$ becomes $8$.