Evaluating ??IncludeSingularTerm
in Mathematica sheds some light:
??IncludeSingularTerm
"IncludeSingularTerm is an option for LerchPhi and Zeta. With
IncludeSingularTerm -> True, terms involving ((k + a)^2)^(-s/2) with
k + a == 0 are included. With IncludeSingularTerm -> False, they are not."
This is also mentioned in the docs of the more general Lerch transcendent, LerchPhi[]
(of which Hurwitz zeta is a special case); unfortunately the docs for Zeta[]
neglect to mention this setting.
Evaluating
Options[Zeta]
{IncludeSingularTerm -> False}
shows that the singular term is excluded by default:
Zeta[2, -1, IncludeSingularTerm -> True]
ComplexInfinity
Zeta[2, -1, IncludeSingularTerm -> False]
1 + ?^2/6