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

Piateskii-Shapiro defined local $L$-factors $L^{PS}(s,\Pi,\Lambda)$ attached to irreducible admissible representations $\Pi$ of the group $GSp(4)$ over local fields and Bessel models of $\Pi$ attached to Bessel data $\Lambda$. 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 $L^{PS}_{ex}(s,\Pi,\Lambda)$ for split Bessel models of $\Pi$.