Estimates of the Levy constant for $\sqrt p$ and class number one criterion for $\mathbb Q(\sqrt p)$

Let $p\equiv3\!\pmod4$ be a prime, let $l(\sqrt p)$ be the length of the period of the expansion of $\sqrt p$ into a continued fraction, and let $h(4p)$ be the class number of the field $\mathbb Q(\sqrt p)$. Our main result is as follows. For $p>91$, $h(4p)=1$ if and only if $l(\sqrt p)>0.56\sqrt p\ L_{4p}(1)$, where $L_{4p}(1)$ is the corresponding Dirichlet series. The proof is based on studying linear relations between convergents of the expansion of $\sqrt p$ into a continued fraction.