Too many Gammas coming in!

Calculus Level 5

\[\large \sum_{r=0}^{\infty} \left(\dfrac{-1}{2}\right)^r \dfrac{\Gamma \left(\frac{1}{2}\right) \Gamma \left(\frac{1+r}{2}\right)}{\Gamma\left(1+\frac{r}{2}\right)}=\dfrac{p_1^{a} \pi^{b}}{p_2^{{c/ }{d}}}\]

The above equation holds true where \(p_1\) and \(p_2\) are primes with integers \(a,b,c\) and \(d\) such that \(c,d\) is coprime.

Find \(p_1+p_2+a+b+c+d\).

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