From what I've found, it's mostly presented as a fait accompli, for instance, the Riemann-Siegel function $Z(t)$, a function frequently used in the zero-finding, is defined with a $\zeta\left(\frac12+it\right)$ factor. What now one looks for when using constructs like these is if the function ever "turns away" from the horizontal axis without crossing it (if this does happen, the hypothesis is false).
This is why things like Lehmer's phenomenon are interesting behavior for the ζ function. Put simply, these are sections of the critical strip where there is (very!) nearly no crossing.
Here are two traditional ways of visualizing the Lehmer phenomenon: one can look at the graph of the Riemann-Siegel function:

(Mathematica code: Plot[RiemannSiegelZ[z], {z, 0, 100}, AspectRatio -> 1/5, Frame -> True]
)
or the so-called "zeta spiral" $z(t)=\zeta\left(\frac12+it\right)$ in the complex plane:

(Mathematica code: ParametricPlot[Through[{Re, Im}[Zeta[1/2 + I t]]], {t, 0, 40}, AspectRatio -> Automatic, Frame -> True]
).
Now, here is the first instance of the Lehmer phenomenon, seen using both viewpoints:

(Mathematica code: Plot[RiemannSiegelZ[z], {z, 7004, 7006}, Frame -> True, PlotRange -> {-2, 2}]
)

(Mathematica code: ParametricPlot[Through[{Re, Im}[Zeta[1/2 + I t]]], {t, 7004 + 1/2, 7005 + 1/2}, AspectRatio -> Automatic, Frame -> True]
)
The hypothesis would have been false if for the Riemann-Siegel function, the function displayed a local extremum without crossing the x-axis, or for the zeta spiral, the apparent "cusp" was actually a cusp or did not even pass through the origin. (I won't spoil the fun by posting zoomed-in versions of those last two images, you can do it yourself with Mathematica or some other computing environment that can evaluate ζ(s) for complex values).
The two zeroes are in fact quite close: FindRoot[RiemannSiegelZ[x], {x, ##}, WorkingPrecision -> 20] & @@@ {{7005 + 1/50, 7005 + 7/100}, {7005 + 9/10, 7005 + 11/100}}
returns the two approximate zeroes 7005.0628661749205932
and 7005.1005646726748389
.