In mathematics, a Barnes zeta function is a generalization of the Riemann zeta function introduced by . It is further generalized by the Shintani zeta function.
The Barnes zeta function is defined by
where w and a<sub>j</sub> have positive real part and s has real part greater than N.
It has a meromorphic continuation to all complex s, whose only singularities are simple poles at s = 1, 2, ..., N. For N = w = a<sub>1</sub> = 1 it is the Riemann zeta function.