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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.
Let ([zeta]) denote the functional equation for the zeta function and (S) the classical sampling theorem; furthermore, let (C) denote the expansion of the cotangent function in partial fractions, (F) a special Fourier series, and (v) the functional equation for Jacobi's theta function.