Inspired by Me - 2

Calculus Level 5

Let

\[S = \left(1- \dfrac{1}{2^4} \right) \left(1- \dfrac{1}{3^4} \right) \left(1- \dfrac{1}{5^4} \right) \left(1- \dfrac{1}{7^4} \right) \ldots\]

If the value of the above product \(S\) can be represented as \(\dfrac{A}{C \pi ^B}\), where \(A\), \(B\) and \(C\) are positive integers and \(A\), \(C\) are coprime to each other, find the value of \(A+B+C\).

Note: The product is taken over all primes, starting from \(2,3,5,7,\ldots\).

You may want to look up the Euler product formula to help you with this problem.

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