number theory - Does this show that the Apery Constant is transcendental?

Last August I posted this on mathoverflow: https://mathoverflow.net/questions/71856/a-serendipitous-riemann-identity. I show the (slightly revised) equation below:

$$\zeta (3)=\frac{2\pi^4}{315} \prod _{n=1}^{\infty } \left(\frac{1}{(p_n){}^2-p_n}+1\right)$$

Since the constant, $\frac{2\pi^4}{315}$ contains $\pi$, which is known to be transcendental, wouldn't this prove that $\zeta(3)$ is transcendental?

I have calculated the product through the first million primes and Mathematica's Element[product,Rationals] returns True. Also, I built a continued fraction of 18,500,045 elements.

The product converges to http://oeis.org/A082695

A paper that uses the product: http://jtnb.cedram.org/cedram-bin/article/JTNB_2004__16_1_107_0.pdf

Answer

No, because an infinite product of rationals is not necessarily rational.

For instance, $$\prod_{n=1}^\infty \left(1-\frac{1}{4n^2}\right)=\frac{2}{\pi}$$

is not rational.