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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