We apply the contour deformation trick to the Thermodynamic Bethe Ansatz equations for the AdS_5 imes S^5 mirror model, and obtain the integral equations determining the energy of two-particle excited states dual to N=4 SYM operators from the sl(2) sector. We show that each state/operator is described by its own set of TBA equations. Moreover, we provide evidence that for each state there are infinitely-many critical values of 't Hooft coupling constant ambda, and the excited states integral equations have to be modified each time one crosses one of those. In particular, the first critical value for the Konishi operator occurs at lambda approx 774. Our results disagree with the ones of Gromov, Kazakov and Vieira, and may explain the mismatch between their recent computation of the scaling dimension of the Konishi operator and the one done by Roiban and Tseytlin by using the string theory sigma model.

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