Dirichlet class number formula, $d>0$: $h(d)=\frac{\sqrt d}{\ln\varepsilon}L(1,\chi)$

Let $d > 0$ be a fundamental discriminant, let $h(d)$ be the number of proper ($\mathrm{SL}_2(\mathbb Z)$-) equivalence classes of binary quadratic forms of discriminant $d$, let $\varepsilon = \frac12(t + u\sqrt d)$ where $t>0$, $u>0$ is the solution of $t^2 - du^2 = 4$ with $u$ minimal, and let $\chi$ be the Dirichlet character modulo $d$ given by $\chi(n) = \left(\frac{d}{n}\right)$. Then

This is Dirichlet's class number formula for real quadratic discriminants.

Formalization Note The source PDF prints $\frac{\sqrt d}{2\ln\varepsilon}L(1,\chi)$. With $h(d)$ and $\varepsilon$ defined as in the source, that constant is off by a factor $2$: for $d=5$ one has $h(5)=1$, $\varepsilon=\frac{3+\sqrt5}{2}$ and $L(1,\chi)=\frac{2}{\sqrt5}\ln\frac{1+\sqrt5}{2}$, so $\frac{\sqrt5}{\ln\varepsilon}L(1,\chi)=1$, while the printed formula gives $\frac12$. The statement here follows Davenport, Multiplicative Number Theory, Ch. 6, the exposition the source says it follows: $h(d)\ln\varepsilon = \sqrt d\,L(1,\chi)$.