A very interesting thing to look at with the "zeta spiral" is Lehmer's phenomenon, which manifests itself as a near-cusp at the origin. Stripping down Clayton's code from above, here's how to visualize the first Lehmer pair:
ParametricPlot[ReIm[Zeta[1/2 + I t]], {t, 7005, 7005 + 1/8},
Background -> RGBColor["#ececeb"], ColorFunctionScaling -> False,
ColorFunction -> Function[{x, y, t},
Blend[{RGBColor["#07617d"],
RGBColor["#f9a828"]},
Norm[{x, y}]/2]],
Frame -> True, PlotRange -> {-0.01, 0.01},
PlotStyle -> Directive[Thickness[0.005], CapForm["Round"]]]

If the spiral did not in fact hit the origin, then the hypothesis would be false there. The MathWorld page I linked to has examples of other Lehmer pairs.
Another nice thing to look at using the spiral would be the Gram points.