It is an incontestable fact that symmetry is one of, if not the, most powerful tools a mathematician can have at his or her disposal. What I want to know, is if there are any good reasons as to why it has been neglected for over a century and a half in the analysis of the zeta function.
In his 1859 paper Riemann proved the functional equation for the zeta function, and defined the function:
$$\xi \left( {s} \right): = \frac{{\Pi \left( {\frac{s}{2}} \right)}}{{{\pi ^{\frac{s}{2}}}}}\left( {s - 1} \right)\zeta \left( s \right) = \frac{{\Pi \left( {\frac{{1 - s}}{2}} \right)}}{{{\pi ^{\frac{{1 - s}}{2}}}}}\left( { - s} \right)\zeta \left( {1 - s} \right)=\xi \left( {1 - s} \right) $$
He then makes the change of variables $$s \mapsto 1/2 + s$$ and redefines the function as
$$\xi (s): = - \frac{{\Pi \left( {\frac{{1/2 + s}}{2}} \right)}}{{{\pi ^{\frac{{1/2 + s}}{2}}}}}\left( {1/2 - s} \right)\zeta \left( {1/2 + s} \right) = - \frac{{\Pi \left( {\frac{{1/2 - s}}{2}} \right)}}{{{\pi ^{\frac{{1/2 - s}}{2}}}}}\left( {1/2 + s} \right)\zeta \left( {1/2 - s} \right) = \xi \left( { - s} \right) $$
And further establishes the product formula
$$\xi \left( s \right) = \xi \left( 0 \right)\prod\limits_{{\mathop{\rm Im}\nolimits} \left( \rho \right) > 0} {\left( {1 - \frac{{{s^2}}}{{{\rho ^2}}}} \right)} $$
which shows that it behaves somewhat like the sine cardinal function, or more generally a Bessel function, except that its roots occur with logarithmic frequency.
The point is that, from the functional viewpoint at the very least, it is very unnatural to use the approach that has been used for the past century and a half for the same exact reason that we do not define, for some a, the function
$$f\left( s \right) = {\mathop{\rm sinc}\nolimits} (s + a) $$
to be the primary definition of the function sinc. So, why would we do that for xi, or by consequence zeta?
Ironically, my own analysis suggests that in neglecting symmetry, the only possible proofs of the original Riemann Hypothesis are through exhaustive methods, which is highly counter-intuitive, but this is way beyond the scope of this post and likely anything that Riemann could have known.