On the values of a zeta function at non-positive integers (Q810082)

Let \({\tilde \zeta}\)(s)\(=\sum_{n_ 1\geq 1,n_ 2\geq 1,n_ 3\geq 0}(n_ 1n_ 2+(n_ 1+n_ 2)n_ 3)^{-s}\) for Re s\(>3/2\). The authors prove that \({\tilde \zeta}\)(s) can be meromorphically continued to the whole complex plane, and evaluate explicitly the values of this zeta-function at non-positive integers in terms of Bernoulli numbers. It turns out that \({\tilde \zeta}\)(-m)\(\in {\mathbb{Q}}\) for \(m\in {\mathbb{Z}}\), \(m\geq 0\).