An investigation of the non-trivial zeros of the Riemann zeta function

While many zeros of the Riemann zeta function are located on the critical line $\mathrm{\Re }(s)=1\mathrm{/}2$, the non-existence of zeros in the remaining part of the critical strip $\mathrm{\Re }(s)\in ]0,1[$ is the main scope to be proven for the Riemann hypothesis. The Riemann zeta functional leads to a relation between the zeros on either sides of the critical line. Given $s$ a complex number and $\bar{s}$ its complex conjugate, if $s$ is a zero of the Riemann zeta function in the critical strip $\mathrm{\Re }(s)\in ]0,1[$, then $\zeta(s) = \zeta(1-\bar{s})$, as a key proposition to prove the Riemann hypothesis.