  We exactly evaluate the third neighbor correlator <S_j^z S_{j+3}^z> and all
the possible non-zero correlators <S^{alpha}_j S^{beta}_{j+1} S^{gamma}_{j+2}
S^{delta}_{j+3}> of the spin-1/2 Heisenberg $XXX$ antiferromagnet in the ground
state without magnetic field. All the correlators are expressed in terms of
certain combinations of logarithm ln2, the Riemann zeta function zeta(3),
zeta(5) with rational coefficients. The results accurately coincide with the
numerical ones obtained by the density-matrix renormalization group method and
the numerical diagonalization.
