  We have computed through order $\beta^{21}$ the high-temperature expansions
for the nearest-neighbor spin correlation function $G(N,\beta)$ of the
classical N-vector model, with general N, on the simple-cubic and on the
body-centered-cubic lattices.
  For this model, also known in quantum field theory as the lattice O(N)
nonlinear sigma model, we have presented in previous papers extended expansions
of the susceptibility, of its second field derivative and of the second moment
of the correlation function.
  Here we study the internal specific energy and the specific heat
$C(N,\beta)$, obtaining new estimates of the critical parameters and therefore
a more accurate direct test of the hyperscaling relation $d \nu(N)=2 -
\alpha(N)$ on a range of values of the spin dimensionality N, including N=0
[the self-avoiding walk model], N=1 [the Ising spin 1/2 model], N=2 [the XY
model], N=3 [the classical Heisenberg model]. By the newly extended series, we
also compute the universal combination of critical amplitudes usually denoted
by $R^+_{\xi}(N)$, in fair agreement with renormalization group estimates.
