In this case, it's worth pointing out that the zeta function admits a functional definition in which you can define some points of the function about their "ref

In this case, it's worth pointing out that the zeta function admits a functional definition in which you can define some points of the function about their "reflection" in the line Re(z) = 1/2. This allows a perfectly rigorous definition of the function for values like -1. One might call the functional form more fundamental and note that the series form only holds true for Re(z) > 1.