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We have computed through order $\beta^{21}$ the high-temperature expansionsfor the nearest-neighbor spin correlation function $G(N,\beta)$ of theclassical N-vector model, with general N, on the simple-cubic and on thebody-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 expansionsof the susceptibility, of its second field derivative and of the second momentof 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 thereforea 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 XYmodel], N=3 [the classical Heisenberg model]. By the newly extended series, wealso compute the universal combination of critical amplitudes usually denotedby $R^+_{\xi}(N)$, in fair agreement with renormalization group estimates.Comment: 15 pages, latex, no figure

Critical specific heats of theN-vector spin models on the simple cubic and bcc lattices