A note on spectral zeta functions of quantum groups

We study analytic properties of spectral zeta functions associated to actions of the quantum group SU_q(2) such as Z(s, SU_q(2)), the zeta function corresponding to the regular representation introduced in [15]. As an application, we show the special value ζ(3) of the Riemann zeta, function ζ(s) is given in terms of the classical limit of Z(s, SU_q(2)). We further discuss a spectral zeta function Z(s, ℂ_q[x, y]) associated with the so-called model of the representations of U_q(s-fraktur sign and l-fraktur sign₂) and show a presence of its series of "trivial" zeros, which is noteworthy.