I am obviously missing something crucial. So this is how wikipedia defines the Riemann Zeta:

And then a few lines down says:

How is this possible? If I put in a zero for each s in the series, I get an infinite series of 1 + 1 + 1 + 1 + ... which must be infinity.

I realize that s can be a complex number... but i'm just not seeing this. Please help.