Does the following product converge as $n$ approaches infinity?

Question

Does the following product converge as $n$ approaches infinity?

$$\lim_{n \to \infty} \prod_{c=2}^n{ \left( 1- \left( 1-\frac{1}{\zeta(c)} \right)^{2^\alpha} \right) }$$

Here $\zeta(c)$ is the Riemann zeta function, and $\alpha$ is a natural.