# Factorials up and down!

**Calculus**Level 5

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

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.