Roots of Digamma

Calculus Level 5

\[\large \sum _{ a\in \Psi }^{ }{ \frac { 1 }{ { a }^{ 3 } } } =-A\zeta \left( B \right) -{ \gamma }^{ C }-\frac { { \gamma \pi }^{ D } }{ E } \]

The above equation, where \(\Psi\) denotes all the roots of digamma function, holds true for positive integers \(A\), \(B\), \(C\), \(D\), and \(E\) with \(B\) being an odd integer. Find \(A+B+C+D+E\).

Notation: \( \gamma\) denotes the Euler-Mascheroni constant, \(\gamma \approx 0.5772 \).

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