As suggested in a comment, here are some plots:
Plot[NIntegrate[Log[Abs[Zeta[a+I t]]],{a,1/2,1}],{t,1,50}]
ParametricPlot[ReIm[NIntegrate[Log[Zeta[a+I t]],{a,1/2,1}]],{t,1,50},PlotPoints->150]
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