[wbl]

You need analytic continuation to define the zeta function at the places you are asking for zeros.

[Y_Y]

That's a good point. I do remember doing problems related to extending formulae outside the radius of convergence in my final year before university, but I don't think it's fair to ask for proper complex analysis from 17-year-olds.

As penance I did go an have a look for suitable numerical techniques for calculating zeta with Re(s)<1 and there are some, e.g. https://people.maths.bris.ac.uk/~fo19175/talks/slides/PGS_ta...