This calculation uses the amazing abilities of SymPy, to evaluate the even-valued zeta functions using Parseval's theorem applied to the functions $t^{s}$.
The last two lines evaluate to
$$2 \sum_{n=1}^\infty \frac{1}{n^2} = \frac{\pi^2}{3},$$
$$\pi^2 \sum_{n=1}^\infty \frac{1}{n^2} + \sum_{n=1}^\infty \frac{1}{n^4} = \frac{8 \pi^4}{45}.$$
Incidentally, this recursive definition provides an easy proof of why all the even $\zeta$s are rational multiples of $\pi^{2s}$.
I wonder if the polynomials in $t$ that evaluate to $\zeta(s)$ are special somehow ...
