The $L$-function of the surface parametrizing cuboids

In this note, we compute the $L$-function of the projective smooth surface $S$ over $\mathbb{Q}$ that parametrizes cuboids whose geometric properties are studied in detail by Stoll and Testa. As a byproduct, we completely determine the structure of $\mathrm{P}\mathrm{i}\mathrm{c}(S_{\bar{\mathbb{Q}}})$ as a $\mathrm{G}\mathrm{a}\mathrm{l}(\bar{\mathbb{Q}}\mathrm{/}\mathbb{Q})$-module.