z = 0 is undefined because 0 is not in the domain of f(z). It's not in the domain because when you plug 0 into the denominator, you get 0 [(0-1)^2 - 1] = 0.

And I'm not sure why you mean by "classify" the zeroes z = pi and z = 2pi. The function doesn't cross the x-axis; it just hits the x-axis at these points and then returns up in the positive y-direction.