Where Did Zeta Come From?

Calculus Level 5

\[\large\sum _{ k=1 }^{ \infty }{ \dfrac { 2k( 2k+1 ) ( 2k+2 ) ( 2k+3 ) \zeta ( 2k+4 ) }{ { 4 }^{ 2k+4 } } } =A-\dfrac { { B\pi }^{ C } }{ D } \]

The above equation holds true for positive integers \(A,B,C\) and \(D\), where \(B\) and \(D\) are coprime. Find \(A+B+C+D\).

Notation: \(\zeta(\cdot) \) denotes the Riemann zeta function.

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