... and yet it clearly states

here that "The Riemann zeta function

may be computed analytically for even n using either contour integration

__or Parseval's theorem with the appropriate Fourier series__" (my underlining). Unfortunately, it doesn't tell you which Fourier series to use.

My guess is that there must be a function whose Fourier coefficients are

. That would get round the problem of having to avoid anything that looks like the Apéry constant, yet still allows you to get

using Parseval's theorem. Have you tried the function

? (That's just a guess — I have no idea whether it will work.)

Actually

(see the above link).