Series $\sum F(m)q^m$, where $F(m)$ is the number of odd classes of binary quadratic forms of determinant $-m$

Consideration of the analytic continuation of the Eisenstein series of weight $3/2$ for the group $\Gamma_0(4)$ leads to a new proof of Mordell's formula connecting the values $\chi(\omega)=\sum^\infty_{m=1}F(m)e^{\pi im\omega}$, $\operatorname{Im}\omega>0$, and $\chi(-\frac{1}{\omega})$. The behavior of the function $\chi(\omega)$for $\Gamma_0(4)$is examined by the same method.