Riemann Zeta function - Zeta(0) != Infinity ??

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

http://upload.wikimedia.org/math/4/2...273e15bd76.png

And then a few lines down says:

http://upload.wikimedia.org/math/0/b...582c54052c.png

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.