For Bloch electrons in a magnetic field, explicit solutions are obtained at the center of the spectrum for the Bethe ansatz equations recently proposed by Wiegmann and Zabrodin. When the magnetic flux per plaquette is 1/Q where Q is an odd integer, distribution of the roots is uniform on the unit circle in the complex plane. For the semi-classical limit, Q → ∞, the wavefunction obeys the power low and is given by |ψ(x)| 2 = (2/ sin πx) which is critical and unnormalizable. For the golden mean flux, the distribution of roots has the exact self-similarity and the distribution function is nowhere differentiable.The corresponding wavefunction also shows a clear self-similar structure.