$C_2 := \prod_{p \geq 3} \big( 1 - (p-1)^{-2} \big)$, where the product ranges over the odd primes $p$.
Comments
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Let $\pi_2(x)$ denote the number of primes $p\leq x$ such that $p+2$ is also a prime. The first Hardy-Littlewood conjecture implies that $\pi_2(x) \sim 2C_2\frac{x}{\log^2 x}$.
