As I found in wikipedia Riemann Hypothesys has been verified numerically by X. Gourdon (2004) up to 10000000000000 ($10^{13}$) zeroes.
I have a few question about how they did it. I tried to read on their official website (numbers.computation.free.fr), but could not find all answers. if I use notation $\zeta(s)$.
Do I correctly understand, that they did not just find zeroes, but they verified Riemann Hypothesis? In other words, if they claim that they do it up to some number (for example $10^{13}$), from scientific point of view it does not make sense to verify it below this value and it does make sense only to go above this number?
Do I correctly understand that they verified up to $10^{13}$ zeros (or up to zero with index $10^{13}$), but not up to Im(s)=$10^{13}$? If yes, how to understand up to which Im(s) they did it?
Is there an explicit formula for zero with index n? What does it mean when they mention "The first column contains a zero index n"?
Do I correctly understand, that to verify it, they first calculated number of zeros like described here and then tried to find all these zeroes and then they checked if Zeta function change sign in Re(s)=1/2. If Zeta function change sign in Re(s)=1/2 with specific Im(s), that means zero can иe only in the point where Re(s)=1/2.