Twin Prime Constant

Question

How would one prove that the twin prime constant $$C_2 = \prod_{p > 2}1-\frac{1}{(p-1)^2} > 0$$ Simply computing the product for a large number of terms isn't rigorous, and simply establishes upper bounds, rather than lower bounds.

Answer 1

$$\prod_{p > 2}\left(1-\frac{1}{(p-1)^2}\right) > \prod_{n \ge 2}\left(1-\frac{1}{n^2}\right) = \frac12$$

Answer 2

$$\prod_{p>2}\left(1-\frac{1}{(p-1)^2}\right)\geq\exp\left(-\frac{6}{5}\sum_{p>2}\frac{1}{(p-1)^2}\right)\geq\exp\left(-\frac{6}{5}(\zeta(2)-5/4)\right).$$