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Proton spin-lattice relaxation times have been measured in the quasi-one-dimensional s = T Heisenberg antiferromagnetic compounds CuSe04 • SH20 and CuS04 • SH20, in order to study the magnetic field dependence of the 1 D quantum spin dynamics at various temperatures. Information is obtained on the low-frequency behavior of the spin autocorrelation functions for the linear chains. The transverse component shows at high Ta marked decrease with magnetic field above the saturation field Be. Towards lower temperatures a pronounced maximum just below Be develops and the decrease above Be becomes more abrupt. The longitudinal component behaves more smoothly near and above Be. The field dependence of the relaxation rates is well explained at all nonzero temperatures by finite-chain calculations. For T =0 we present analytical calculations of autocorrelation functions based on the predominance of a two parameter spin-wave continuum for the quantum spin dynamics yielding a non divergent behavior at w = 0 and B = 0. Exact results are obtained for fields above Be. Further, an effective XY approach is used for low but finite T. Here, good agreement is found with the proton relaxation data for very low temperatures.