Regular poles for spinor $L$-series attached to split Bessel models of $\mathrm{G}\mathrm{S}\mathrm{p}(4)$

For irreducible smooth representations $\mathrm{\Pi }$ of $\mathrm{G}\mathrm{S}\mathrm{p}(4,k)$ over a non-archimedean local field $k$, Piatetskii-Shapiro and Soudry have constructed an $L$-factor depending on the choice of a Bessel model. It factorizes into a regular part and an exceptional part. We determine the regular part for the case of split Bessel models.