We consider operators on compact closed manifolds, with coefficients, first derivatives of which are continuous with continuity modulus and derive semiclassical spectral asymptotics with sharp remainder estimate O(h1−d); for operators with continuity modulus we derive semiclassical spectral asymptotics with the remainder estimate o(h1−d) under standard condition to Hamiltonian flow. For operators with even less regular coefficients we establish less sharp spectral asymptotics. These asymptotics easily yield standard asymptotics with respect to spectral parameter . We will treat operators on manifolds with boundary in the next paper.