The spherical Hall algebra of $\overline{\mathrm{Spec}({\mathcal{O}}_K)}$

We generalize a result of M. Kapranov, O. Schiffmann, and E. Vasserot by showing that, for a number field $K$ with class number one, the spherical Hall algebra of $\overline{\mathrm{Spec}({\mathcal{O}}_K)}$, where ${\mathcal{O}}_K$ is the ring of integers of $K$, is isomorphic to the Paley-Wiener shuffle algebra associated to a Hecke $L$-function corresponding to $K$.