On single-variable Witten zeta functions of rank 2 and 3

Abstract: We employ a Mellin transform-based approach to simultaneously study single-variable Witten zeta functions associated with rank 2 and 3 root systems. Detailed information about their pole locations, residues, and special values are obtained. Regarding their behavior at negative integers, we highlight a potential connection to Eisenstein series and make a $p$-adic observation.