Just a thought that crossed my mind after reading about the Riemann hypothesis.

Riemann expanded the domain of the Zeta function towards all complex numbers except for the pole at 1. Then, after calculating the zero's for this function, he landed on the hypothesis about the non-trivial zero's all residing on the line with real part = 1/2.

These non-trivial zero's are all of the shape 1/2 +/- y i. Since they come in pairs, these imaginary points could be zero's of a real kwadratic function of the shape a^2 X + b X + c with (b^2 - 4ac) < 0. Injecting the non-trivial zero's of the shape 1/2 +/- y i would induce a set of functions of the shape:

X^2 - X + y^2 + 1/4

This of course doesn't reveal any better pattern than the complex non-trivial zeros, but I wondered whether this could imply that a real function that produces all the non-trivial zeros exists?