\sum_{\substack{r|\nu }}
\frac{\epsilon_p(r)}{r} \delta_{-m,p\nu\nu'/r^2}{\mathcal{K}}_{s}(2\pi \nu y_1)
{\mathcal{K}}_{s}(2\pi \nu' y_2)e(\nu x_1+\nu' x_2)\\
&\phantom{=}{}+2\sqrt{p} \sum_{\substack{ \lambda\in\partial_F^{-1}\\ {\operatorname{N}}(\lambda)=-m/p}} \sum_{n\geq 1}
\frac{\epsilon_p(n)}{n}
 {\mathcal{I}}_{s}(2\pi n \beta(\lambda y_1,\lambda' y_2))
{\mathcal{K}}_{s}(2\pi n \alpha(\lambda y_1,\lambda' y_2)) e(n\lambda x_1+n\lambda' x_2).\end{aligned}$$ By rearranging the sums one deduces the stated formula.

Now it can be proved as in [@Br1] that $\tilde\Phi_{mp^2}(z_1,z_2,s)$ has a meromorphic continuation in $s$ to a neighborhood of $s=1$. The continuation turns out to be holomorphic at $s=1$, whereas there always was a simple pole in [@Br1]. This follows from the presence of $L(2s-1,\epsilon_p)$ in the constant term of the Fourier expansion given in Theorem \[fourier1\], while there appeared $\zeta(2s-1)$ in [@Br1] Theorem 1.

We define the regularized Green function $\tilde\Phi_{mp^2}(z_1,z_2)$ for $\tilde T_m$ as the value of $\tilde\Phi_{mp^2}(z_1,z_2,s)$ at $s=1$.

One finds that $\tilde\Phi_{mp^2}(z_1,z_2)$ is a harmonic function on ${\mathbb{H}}^2\setminus\tilde T_m$ with a logarithmic singularity along $-2\tilde T_m$.

\[fourier2\] The Green function $\tilde\Phi_{mp^2}(z_1,z_2)$ associated to $\tilde T_m$ has the Fourier expansion $$\begin{aligned}
&\tilde\Phi_{mp^2}(z_1,z_2)\\
&=\sqrt{p}L(1,\epsilon_p)
b_m(0,1)\\
&\phantom{=}{}+\sqrt{p} \sum_{\substack{\nu\in \partial_F^{-1}\\\nu\nu'> 0}}\sum_{n\geq 1}
 \frac{\epsilon_p(n)}{n} b_m(p\nu\nu',1)e(-2\pi n|\nu y_1+\nu' y_2|)e(\nu n x_1+\nu'n x_2)\\
&\phantom{=}{}+\sqrt{p} \sum_{\substack{ \lambda\in\partial_F^{-1}\\ {\operatorname{N}}(\lambda)=-m/p}} \sum_{n\geq 1}
\frac{\epsilon_p(n)}{n} e\big(-2\pi n |\lambda y_1+\lambda' y_2|\big) e(n\lambda x_1+n\lambda' x_2).\end{aligned}$$ It converges normally on $y_1 y_2>mp$ outside the polar part of $\tilde T_m$.

This 