BUG: zeta(1/2, 1) returns nan

Describe your issue.

The Hurwitz zeta function at 1/2, 1 is equivalent to Riemann zeta function at 1/2 and so zeta(1/2, 1) = zeta(1/2) = -1.46... (See WolframAlpha) but scipy.special.zeta returns nan. This may be expected behavior but if so it would be nice to have documentation for this behavior.

Related issue: https://github.com/pytorch/pytorch/issues/68110

Reproducing Code Example

from scipy.special import zeta

# correct
output = zeta(1 / 2)
print(output)

# wrong
output = zeta(1 / 2, 1)
print(output)

Error message

-1.4603545088095866            
nan

SciPy/NumPy/Python version information

1.7.1 1.21.2 sys.version_info(major=3, minor=9, micro=8, releaselevel=‘final’, serial=0)

[simaki]

Thank you!

In my understanding, while the original definition of Hurwitz zeta zeta(s,a) only works for Re(s)>1 as you pointed out, one would be able to further define it for s!=1 through analytic continuation (cf https://dlmf.nist.gov/25.11). So zeta(s,1)=zeta(s) would hold for s!=1.

On the other hand I personally presume even if the statement above is mathematically true the implementation for Re(s)<1 may not be really necessary for a moment because its use case would be rare. Actually I just happened to find it when I was examining https://github.com/pytorch/pytorch/issues/68110

[Gattocrucco]

Right now I need special.zeta(s, a) with negative s. I’m using the one from JAX which implements it. Edit: JAX supports only s > 1 too (https://github.com/google/jax/issues/11521). The function is meromorphic, so it’s well defined, there are no branch cuts, and it’s real for real arguments.