This is why mathematicians long to prove the hypothesis. Researchers have used supercomputers to calculate the first 1,500,000,001 zeros above the x-axis, and millions of other zeros higher up, and so far all of them lie on the critical line. If just one of them did not, the Riemann hypothesis would be killed. This is heartening, but no amount of computer hacking can prove the hypothesis. There are always more zeros to check.
Some deeper insight is needed.
Quantum mechanics deals with the behaviour of tiny particles such as electrons. Crucially, its equations work with complex numbers, but the energy of a physical system is always measured by a real number. This sounds like Riemann's zeros. The line of zeros is vertical, rather than horizontal, but it is a simple bit of maths to rotate it and put it on top of the real line. If the zeros then match up with the energy levels of a quantum system, the Riemann hypothesis is proved.