An algebra of pseudodifferential operators on an arbitrary unimodular Lie group is constructed. It is defined by uniform estimates of the local symbols and by some conditions on the decreasing of the Schwartz kernel, which are formulated by means of a weight function that increases intermediately between the volume function and the standard exponential function. The decreasing of the Green function is described, complex powers of elliptic operators from the algebra are constructed. The meromorphic continuation of the Schwartz kernel of the complex powers is described, which implies an asymptotic of the spectral function of the operators.