zeta(2N)

Euler determined the values of the Riemann zeta function for positive even integers in 1734 by considering the Maclaurin series for the sine function.  He proved that

and found more values of the zeta function for positive even integers.  The fact that π is involved in these expressions was rather unexpected.  These neat results are not shared, or at least not known to be shared, by the zeta function values for positive odd integers.