  In this paper we perform a detailed analysis of Riemann's hypothesis, dealing
with the zeros of the analytically-extended zeta function. We use the
functional equation $\zeta(s) = 2^{s}\pi^{s-1}\sin{(\displaystyle \pi
s/2)}\Gamma(1-s)\zeta(1-s)$ for complex numbers $s$ such that $0<{\rm Re(s)}<1$
and the reduction to the absurd method where we use an analytical study based
on a complex function and its modulus as a real function of two real variables
in combination with a deep numerical analysis to show that the real part of the
non-trivial zeros of the Riemann zeta function is equal to $1/2$ to the best of
our resources. This is done in two steps. Firstly, we show what would happen if
we assumed that the real part of $s$ has a value between $0$ and $1$ but
different from $1/2$ arriving at a possible contradiction for the zeros.
Secondly assuming that there is no real value $y$ such that $\zeta\left(1/2 +yi
\right)=0$ by applying the rules of logic to negate a quantifier and the
corresponding Morgan's law we also arrive to a plausible contradiction.
Finally, we analyze what conditions should be satisfied by $y \in \mathbb R$
such that $\zeta(\displaystyle 1/2 +yi)=0$. While these results are valid to
the best of our numerical calculations, we do not observe and foresee any
tendency for a change. Our findings open the way towards assessing the validity
of Riemman's hypothesis from a fresh and new mathematical perspective.
