It is known that Apéry's constant defined as $$\zeta (3)=\sum _{n=1}^{\infty }\frac {1}{n^3}= \frac{5}{2}\sum _{{n=1}}^{\infty }{\frac {(-1)^{{n-1}}}{n^{3}{\binom{2n}{n}}}}=1.2020569\cdots$$ was proven to be irrational by Roger Apéry in 1979.
But is $\zeta^2(3)= 1.44494079\cdots$ also irrational?
I have input this value into Wolfram Alpha, and the program cannot tell if the value is rational or not, so I assume the rationality of $\zeta^2(3)$ is unknown. However, I do not quite understand the difficulty of proving the rationality of $\zeta^2(3)$ since $\zeta (3)$ is already proven irrational. I know that $\pi^2$ and $e^2$ are irrational, but why is it hard to prove the rationality of $\zeta^2(3)$?