In this paper, a study is made of the properties of the energy E(k) of the Hill operator, as a function of the "global" quasimomentum k, running over the Riemann surface of the quasimomentum K. A study is made of Ė, Ë on the "straight line" Im k=0, and of the location of the zeros of Ë. For example: 1) the function Ė on the straight line Im k=0 between zeros has only one local maximum or minimum, 2) in the case of an N-zone potential, the function Ë has exactly 6(N-1) zeros in K, N-1 zeros both on the positive (negative) "semiaxes," and in each quadrant.