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The right-hand side in https://dlmf.nist.gov/25.11#E35 looks like a perfectly fine global definition of the alternating Hurwitz zeta function, with the exception of special points. The restrictions Re(a) > 0 and Re(s) > 0 are just there to make the sum and integral well-defined.

The special points are s = 1 (where you need to compute the limit 0.5 * (digamma((a+1)/2) - digamma(a/2))) and the points where either Hurwitz zeta function is not defined with respect to a, namely nonnegative integer a (except when s is a nonnegative integer too).

The right-hand side in https://dlmf.nist.gov/25.11#E35 looks like a perfectly fine global definition of the alternating Hurwitz zeta function, with the exception of special points. The restrictions Re(a) > 0 and Re(s) > 0 are just there to make the sum and integral well-defined.

The special points are s = 1 (where you need to compute the limit 0.5 * (digamma((a+1)/2) - digamma(a/2))) and the points where either Hurwitz zeta function is not defined with respect to a, namely nonnegative nonpositive integer a (except when s is a nonnegative nonpositive integer too).