$$\frac{1}{\frac{\zeta (2 k)}{\pi ^{2 k}}}$$
Consider the sequence of values generated for the above for increasing values of $k$. The sequence starts like $6,90,945,9450,93555,\frac{638512875}{691},\dots$
Now plot the values of this sequence. They appear exponential in nature. Log-ing the graph confirms this.
Now naturally one would want to do a regression on the data. Doing so with values up to $k=2000$ produces $0.99999e^{2.28946017}$. The larger $k$ gets, the closer the coefficient gets to $1$.
So several questions naturally arise from this experiment.
A) Why does this sequence have an exponential distribution.
B) What's significant about that $2.28946017$.
From the definition of $\zeta(s)$ we see that $\zeta(k)\to 1$ as $k\to\infty$. This implies $$ \frac{1}{\zeta(2k)/\pi^{2k}}\sim \pi^{2k}= 1\cdot e^{(2\log\pi)k}. $$ We have $2\log\pi\approx 2.28945977$.