400 Followers Problem - Reciprocal Summations 3

Calculus Level 5

$\large{S = 1 + \dfrac{1}{5}- \dfrac{1}{7} - \dfrac{1}{11} + \dfrac{1}{13} + \dfrac{1}{17} - \dfrac{1}{19} - \dfrac{1}{23} + \ldots }$

If $$S$$ can be expressed as $$\dfrac{\pi^A}{B}$$ for positive integers $$A,B$$, submit the value of $$A+B$$ as your answer.

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