Sur la fonction Z\^eta associ\'ee au Laplacien singulier ${\mathrm{\Delta }}_{{\mathcal{O}(m)}_{\mathrm{\infty }}}$

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 ${\zeta }_{\mathrm{\infty }}$, 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 ${\zeta }_{\mathrm{\infty }}(0)$ and ${\zeta }_{\mathrm{\infty }}^{\mathrm{\prime }}(0)$.