The antiferromagnetic and superconducting properties of a thiospinel ${\mathrm{Cu}}_{1.5}$${\mathrm{Co}}_{1.5}$${\mathrm{S}}_{4}$ (${\mathit{T}}_{\mathit{N}}$=19.0 K, ${\mathit{T}}_{\mathit{c}}$=2.3 K) have been investigated with $^{63}\mathrm{Cu}$ NMR (75 MHz), $^{59}\mathrm{Co}$ NMR (75 MHz), and $^{59}\mathrm{Co}$ pure quadrupole resonance between T=1.23 and 150 K. The linear dependence of negative $^{63}\mathrm{Cu}$ Knight shift (-0.013% at 4.2 K) on the Curie-Weiss-type susceptibility \ensuremath{\chi}(T) and nearly independent $^{59}\mathrm{Co}$ Knight shift (+1.43%) indicate that the d hole band of Cu at the tetrahedral A site is mainly responsible for the spin paramagnetism. The spin-lattice relaxation rates (${\mathit{T}}_{1}$T${)}^{\mathrm{\ensuremath{-}}1}$ of both $^{63}\mathrm{Cu}$ and $^{59}\mathrm{Co}$ are significantly enhanced with lowering temperature below \ensuremath{\sim}100 K, similar to those observed in high-${\mathit{T}}_{\mathit{c}}$ copper oxygen perovskite superconductors, which are associated with the growth of antiferromagntic spin correlations at low temperatures. Below ${\mathit{T}}_{\mathit{c}}$, ${\mathit{T}}_{1}^{\mathrm{\ensuremath{-}}1}$ of the fractional Co at the octahedral B site, which increases from \ensuremath{\sim}0% at 0.65${\mathit{T}}_{\mathit{c}}$ to \ensuremath{\sim}30% at 0.5${\mathit{T}}_{\mathit{c}}$, shows a rapid decrease, indicating a partial formation of the superconducting energy gap. On the other hand, ${\mathit{T}}_{1}^{\mathrm{\ensuremath{-}}1}$ of the dominant part of Co follows a Korringa-like relation down to 0.5${\mathit{T}}_{\mathit{c}}$, suggesting it is in a gapless superconducting state, probably due to strong antiferromagnetic spin correlations.
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