Non trivial zeros of the Zeta function using the differential equations

Using the differential equations, we obtain a more flexible expression for the Riemann Zeta function on the critical strip. This allows us to prove that for every $\tau\in \mathbb{R}^*$ there exists at most a unique point $r\in (0,1)$ such that $\Im\Big(\zeta(r+i\tau)\Gamma(r+i\tau) \Big)=0$, where $\Gamma$ is the Gamma function.