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:


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