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WhatistheAsociated sum series of the Euler Product ofover the Twin Primes?

Please consider the (presumably infinite) Euler product over the twin primes:

$$f(z) = \prod_{p\in\mathbb{P}}^{\infty} \Big( 1 - \frac{1}{p(p+2)} frac{1}{(p(p+2))^ z} \Big)$$ (in which $p(p+2)$ is a divisor of $4((p-1)!+1) + p$ ).

The Euler Product is a product of a corresponding Dirichlet series. Which one is that?

Max

Eddit

Edit Update: $\textbf{please look at first my comment}$ to overcome my lack of knowledge of Latex to improve the questionerror fixed.

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Please consider the (presumably infinite) Euler product over the twin primes:

$$f(z) = \prod_{p\in\mathbb{P}}^{\infty} \Big( 1 - \frac{1}{p(p+2)} \Big)$$ (in which $p(p+2)$ is a divisor of $4((p-1)!+1) + p$ ).

The Euler Product is a product of a corresponding Dirichlet series. Which one is that?

Eddit: $\textbf{please look at first my comment}$ to overcome my lack of knowledge of Latex to improve the question.