I made a small matlab script that finds the zeroes in Riemann zeta function using Newton's method and I though if it would be possible to somehow prove it by showing that Newton's method will always converge to the trivial zeroes or to the Re(s) = 1/2 line.
Now, I'm not a mathematician, I'm an undergraduate in electrical engineering, so I don't fully know complex analysis with exception of a thing here or there so I don't know if this is feasible somehow, I just had this idea and though that it could be helpful or a nice insight.