Exceptional poles of local $L$-functions for $GSp(4)$ with respect to split Bessel models

Piateskii-Shapiro defined local $L$-factors $L^{PS}(s,\mathrm{\Pi },\mathrm{\Lambda })$ attached to irreducible admissible representations $\mathrm{\Pi }$ of the group $GSp(4)$ over local fields and Bessel models of $\mathrm{\Pi }$ attached to Bessel data $\mathrm{\Lambda }$. These local $L$-factors decompose into a product $L^{PS}(s,\Pi,\Lambda)= L^{PS}{ex}(s,\Pi,\Lambda) L^{PS}{reg}(s,\Pi,\Lambda)$ of an exceptional and a regular $L$-factor. In this paper we compute the exceptional factors $L_{ex}^{PS}(s,\mathrm{\Pi },\mathrm{\Lambda })$ for split Bessel models of $\mathrm{\Pi }$.