For Re(k)>-3, the average of zeta'(1/2+i gamma)^k over zeros is conjectured to be (1/Gamma(k+2)) (log(T/2pi))^k.
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Complex moments of the derivative of the Riemann zeta function
For Re(k)>-3, the average of zeta'(1/2+i gamma)^k over zeros is conjectured to be (1/Gamma(k+2)) (log(T/2pi))^k.