The dynamic structure factor S(k,t) as a function of the magnitude k of the scattering vector and time t and its first cumulant Ohm(k) are evaluated in the Gaussian approximation the basis of the coarse-grained helical wormlike chain model. In this approximation, S(k,t) and therefore Ohm(k) may be expressed in terms of the solutions of the 1(1) eigenvalue problem previously obtained. It is then found that the eigenvalues in the j=0 branch of the 1(1) eigenvalue spectrum at small wave numbers make main contribution to Ohm(k). The numerical results for typical flexible polymers show that the so-called "universal" plot of eta(0) Ohm(k)/k(B)Tk(3) against [S-2](1/2)k depends on the kind of polymer, where eta(0) is the solvent viscosity, k(B) the Boltzmann constant, T the absolute temperature, and [S-2] the mean-square radius of gyration. It is also shown that for semiflexible chains, the plot depends appreciably on the reduced total contour length. A comparison of theory with experiment is made with respect to Ohm(k) and also the coefficient C introduced by Stockmayer and co-workers. It is then found that the theory may give a rather satisfactory explanation of experimental results for both flexible and semiflexible polymers despite the fact that the preaveraged Oseen tensor is used along with the Gaussian approximation.