The distribution of Bethe roots, solution of the inhomogeneous Bethe equations, which characterize the ground state of the periodic XXX Heisenberg spin-$frac{1}{2}$ chain is investigated. Numerical calculations shows that, for this state, the new inhomogeneous term does not contribute to the Baxter T-Q equation in the thermodynamic limit. Different families of Bethe roots are identified and their large N behaviour are conjectured and validated.
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