As suggested in a comment, here are some plots:

    Plot[NIntegrate[Log[Abs[Zeta[a+I t]]],{a,1/2,1}],{t,1,50}]

[![enter image description here][1]][1]


    ParametricPlot[ReIm[NIntegrate[Log[Zeta[a+I t]],{a,1/2,1}]],{t,1,50},PlotPoints->150]

[![enter image description here][2]][2]


  [1]: https://i.sstatic.net/hwqAx.png
  [2]: https://i.sstatic.net/ZSD0M.png