Finding roots of infinite series?

I find that hard to believe. The non-trivial zeros of the Riemann zeta function occur in the critical strip, where the real part of z has 0 < Re(z) < 1. The sum you've written diverges whenever the real part is <= 1. The trivial zeros of the zeta function occur at negative even integers with 0 imaginary part. Your question is essentially "prove the Riemann hypothesis"--either a joke or trolling, but I can't quite tell which.