We numerically calculate the spatial correlators of the scalar and pseudoscalar operators F² and F, in SU(3) Yang-Mills theory at zero and finite-temperature on the lattice. We compare the results over the distances 1/2T < r < 3/2T to the free-field prediction, to the operator-product expansion as well as to the strongly coupled large-N_c = 4 super-Yang-Mills theory, where results are obtained by AdS/CFT methods. For T_c < T < 1.15T_c, both channels exhibit stronger spatial correlations than in the vacuum, and we give an explanation for this, using sum-rules and the operator-product expansion. The AdS/CFT calculation provides a semi-quantitatively successful description of the vacuum-subtracted F² correlator, renormalized in the 3-loop scheme, in the interval of temperatures 1.2 < T/T_c < 1.9, while the free-field prediction has the wrong sign. The F and F² correlators are predicted to have the same functional form both at weak coupling and in the strongly coupled SYM theory. The Yang-Mills plasma does not meet that expectation below 2T_c. Instead we find that strong fluctuations of F are present at least up to that temperature. We discuss the impact of our results on our understanding of the quark-gluon plasma.