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 $8$.