I found interesting results when I put in certain values for z into zeta(z). I believe my findings do not disprove the Riemann hypothesis, but I am not so sure.
When I input z = 1/3 + i into the function, I got a result with a positive real component and a negative imaginary component, which can be written as a - bi.
When I input z = 1/3 - i, I got a result with a positive real component and a positive imaginary component, which can be written as c + di.
For z = 2/5 - i/2, I got -f + gi.
For z = 2/5 + i/2, I got -h - li.
Because the Riemann Zeta Function is continuous, these results imply that there is a number in this range z for which zeta(z) has a real component of zero. In fact, there would be infinitely many of these numbers in an unbroken chain. It also implies that there is a number in this range z for which zeta(z) has a complex component of zero and there there are an infinitely many of these in a broken chain. Also, these two chains should meet with each other at some point. This seems like it would disprove the Riemann Hypothesis.