Finite form of $L(1,\chi)$ for prime conductor $q\equiv1\pmod 4$

Let $q$ be a prime with $q\equiv 1\pmod 4$, and let $\chi$ be the Dirichlet character modulo $q$ given by the Legendre symbol, $\chi(m) = \left(\frac{m}{q}\right)$. Then

Combined with the case $d = q$ of Dirichlet's formula, this gives a finite expression for the class number in terms of logarithms of sines.

Formalization Note The source PDF prints the constant $-\frac{1}{2q^{1/2}}$. With the sum over all $m=1,\dots,q-1$ that constant is off by a factor $2$: for $q=5$, $L(1,\chi)=\frac{2}{\sqrt5}\ln\frac{1+\sqrt5}{2}\approx0.4304$, while $-\frac{1}{\sqrt5}\sum_{m=1}^{4}\left(\frac m5\right)\ln\sin\frac{m\pi}{5}\approx0.4304$ and the printed constant gives half of this. The statement here uses $-\frac{1}{q^{1/2}}$. In the source, $\chi = \left(\frac{d}{\cdot}\right)$ with $d = q$, which equals $\left(\frac{m}{q}\right)$ by quadratic reciprocity.