On the non homogeneous quadratic Bessel zeta function

We study the non homogeneous quadratic Bessel zeta function $\zeta_{RB}(s,\nu,a)$ defined as the sum of the square of the positive zeros of the Bessel function $J_\nu(z)$ plus a positive constant. In particular, we give explicit formulas for the main associated zeta invariants, namely poles and residua, $\zeta_{RB}(0,\nu,a)$ and $\zeta'_{RB}(0,\nu,a)$.