Let $K$ be an imaginary quadratic field and $D_K$ its discriminant. Further let $\mathcal O$ be an order in $K$ with conductor $f$ and

$$L(\chi,s)=\sum_{\mathfrak a}\chi(\mathfrak a)N(\mathfrak a)^{-s}$$

where the sum is over all integral ideals of $\mathcal O$ relatively prime to $f$, and $\chi$ is a character of the ideal group of $\mathcal O$ (ideals taken prime to $f$).

In his paper Zum Beweise des Starkschen Satzes, Siegel deduces the formula

$$\frac {1}{2\pi}f |D_K|^{1/2}L(\chi, 1)=-\sum_{\mathfrak C}\bar\chi(\mathfrak c)\log \left( \sqrt{\Im(\omega_{\mathfrak c})\eta(\omega_{\mathfrak c})^2}\right).$$ Here the sum on the right is over ideal classes in $\mathcal O$ (again prime to $f$), $\eta$ is the dedekind eta function, and $\omega_\mathfrak c=-\beta/\alpha$ if $\mathfrak c= [\alpha,\beta]$. This should follow from the Kronecker's first limit formula. Is there some English reference for the Siegel's formula, and the mentioned $L$-functions for non-maximal orders?