  We consider short range correlations in excited states of the finite XXZ and
XXX Heisenberg spin chains. We conjecture that the known results for the
factorized ground state correlations can be applied to the excited states too,
if the so-called physical part of the construction is changed appropriately.
For the ground state we derive simple algebraic expressions for the physical
part; the formulas only use the ground state Bethe roots as an input. We
conjecture that the same formulas can be applied to the excited states as well,
if the exact Bethe roots of the excited states are used instead. In the XXZ
chain the results are expected to be valid for all states (except certain
singular cases where regularization is needed), whereas in the XXX case they
only apply to singlet states or group invariant operators. Our conjectures are
tested against numerical data from exact diagonalization and coordinate Bethe
Ansatz calculations, and perfect agreement is found in all cases. In the XXX
case we also derive a new result for the nearest-neighbour correlator
$\{\sigma_1^z\sigma_2^z\}$, which is valid for non-singlet states as well. Our
results build a bridge between the known theory of factorized correlations, and
the recently conjectured TBA-like description for the building blocks of the
construction.
