For a wide class of infinitely divisible laws μ numbers R(μ) are evaluated satisfying the property: if R>R(μ) then μ is uniquely determined by its values on the half-line (−∞, R); if R<R(μ) then μ is not determined by its values on (−∞, R). Necessary and sufficient conditions are given for ‘half-line’ convergence to μ on (−∞, R) to be equivalent to weak convergence to μ.