  In a previous paper, we computed the spectrum of the singular Laplacian
attached to canonical metrics on $\mathbb{P}^1$. In this article, we study
$\zeta_\infty$, the Zeta function associated to this spectrum. We prove that it
admits a meromorphic continuation to $\mathbb{C}$. Moreover, this continuation
is holomorphic at zero. We compute $\zeta_\infty(0)$ and $\zeta'_\infty(0)$.
