Factorials up and down!

$\begin{aligned} &\sum_{n=1}^\infty\dfrac{\Gamma\left(n+\frac{1}{2}\right)}{(2n+1)^4\,4^n\,n!} =\sqrt{\pi}\left(\dfrac{\pi}{A}\zeta(B)+\dfrac{C}{D\sqrt{E}}\psi^{(F)}\left(\dfrac{G}{H}\right)-\dfrac{\pi^I}{J\sqrt{K}}-L\right) \end{aligned}$

The above equation is true for positive integers $A,B,C,D,E,F,G,H,I,J,K$ and $L$ .

Find the minimum value of: $A+B+C+D+E+F+G+H+I+J+K+L$ .

Notations :

• $\Gamma(\cdot)$ denotes the gamma function .
• $\zeta(\cdot)$ denotes the Riemann zeta function .
• $\psi^{(n)}(\cdot)$ denotes the $n^\text{th}$ derivative of the digamma function .

---

This question is taken from another forum. I found it interesting and hence, decided to share it here.