I know that there are zeros to the zeta function like:
14.134, 21.022, 25.010, etc.

Now here is the problem:

I know about a dozen different functional equations of the zeta function, but they all involve zeta(s), for example:

zeta(1-s)=[2(zeta(s))(gamma(s))cos((pi)s/2)]/((2(pi))^s)

Now I thought that zeta(s) diverges for any real part less than or equal to one, so how is one supposed to solve for zeta(s) in anything when you have to usually compute the zeta of something else as well that diverges.

Basically, where do values like 14.134 come from and what equation is used such that I can plug in s=.5+14.134i and get zero?