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Riemann zeta function for nonnegative even integers. The Riemann zeta function for negative even integers is 0 (those are the trivial zeros of the Riemann zeta function). The Riemann zeta function for nonnegative even integers is given by (note that is the only rational number for nonnegative even integers) where are the Bernoulli numbers.
The answer is in the following product taken over the primes p (also discovered by Euler): Euler wrote this as. Riemann later extended the definition of (s) to all complex numbers s (except the simple pole at s=1 with residue one). Euler's product still holds if the real part of s is greater than one. Riemann derived the functional equation of the Riemann zeta function: