In the paper [*Zum Beweise des Starkschen Satzes*](https://eudml.org/doc/141914) Siegel considers the function


$$L\_q(s)=\sum\_{n=1}^{\infty}\left(\frac{q}{n}\right)n^{-s},$$


where $q$ is a discriminant of a quadratic number field and the character is the Kronecker symbol. Then he writes that "according to Dirichlet" we have, in case $G>0$,


$$L\_G(1)=2G^{-1/2}h\_G\log \varepsilon\_G,$$


where $h\_G$ is the corresponding class number and $\varepsilon\_G$ the fundamental unit. 


However, according to the book [Zetafunktionen und quadratische Körper](https://link.springer.com/book/10.1007/978-3-642-61829-1) the formula reads


$$h\_G=\frac{G^{1/2}}{\log \varepsilon\_G}L\_G(1).$$


Which one is correct? Am I making some mistake?



Is this Siegel's formula correct?
Dirichlet proved his class number formula for quadratic forms; in particular he was working with class numbers $h^+$ in the strict sense, and his unit $\varepsilon$ was the fundamental solution of the Pell equation $t^2 - Du^2 = 1$, not the fundamental unit $\eta$ of the corresponding number field. The relation
$$ \eta^{2h} = \varepsilon^{h^+} $$ 
encodes the two cases


* $\varepsilon = \eta$, $h^+ = 2h$
* $\varepsilon = \eta^2$, $h^+ = h$.


