A simple observation of the behavior of $\zeta(s), s=\sigma +it$ that I wonder if there's a explanation for:
Take $t_k$ as the height of the $kth$ non-trivial zero $z_k$ on the critical line ($\sigma=0.5$), i.e.,
$z_k=0.5+t_k i$
$\zeta(z_k)=0$.
Now consider $z'_k$ a short distance $\epsilon$ away parallel to the real axis, i.e.,
$z'_k= z_k+\epsilon$
$\epsilon \in \mathbb{R}, 0 < \epsilon \ll 1$
The observation has to do with the orientation of $\zeta(z'_k)$ and $\zeta(1-z'_k)$ in the complex plane.
E.g., for $t_{100}=236.524229...$ and $\epsilon=.001$ (origin of complex plane at top center):
Looking at a number of such cases, it would appear that in the horizontal vicinity of a zero on the critical line ($\epsilon$ small), $\zeta(s)$ and $\zeta(1-s)$ approach being equal in magnitude and reflected around the imaginary axis, i.e., the bisector between them is oriented either at $+\pi/2$ or $-\pi/2$.
Does this reflection around the imaginary axis follow from the functional equation?
EDIT:
As an illustration of what's happening, here's an animation:
This equality of angles holds for any values of a complex variable in a movable coordinate system that is rotated by an angle of $-Arg(\chi(s))/2$.
It is easy to explain if we consider the argument equation of the functional equation of the Riemann zeta function:
$Arg(\zeta(s))=Arg(\chi(s))+Arg(\zeta(1-s))$
If we substitute
$Arg(\zeta(\overline{1-s}))=Arg(\overline{\zeta(1-s)})=-Arg(\zeta(1-s))$
and subtract from both parts of the equation $Arg(\chi(s))/2$ (which corresponds to a rotation by an angle of $-Arg (\chi(s))/2$), we obtain:
$Arg(\zeta(s))-Arg(\chi(s))/2=-(Arg(\zeta(\overline{1-s})-Arg(\chi(s))/2)$
By the way it is the key to the Riemann hypothesis,
because when $\sigma<1/2$ the vectors $\zeta(s)$ and $\zeta(\overline{1-s})$ rotate in a moving coordinate system, and is alternately aligned with the axes of this moving coordinate system, and therefore the projection of the vectors $\zeta(s)$ and $\zeta(\overline{1-s})$ (each of them separately) on these axes cannot be zero simultaneously, since these projections are conjugate harmonic functions, that define the Riemann zeta function, then Riemann zeta function cannot have zeros with $\sigma<1/2$ and $\sigma>1/2$.