The authors consider the dynamics of the n-component Ginzburg-Landau model with nonconserved order parameter (model A) following a quench from a high-temperature equilibrium state to zero temperature. The two-time correlation function of the order-parameter field is found in the 1/n expansion to have the asymptotic scaling form Ck(t,t')=t'2/(t/t')2/f (kt,kt') for t>>t', with f(0,0)=constant. The form of the new exponent lambda (which is a non-trivial function of n and d) was given explicitly to O(1/n) in a recent letter. The purpose of this study is to present a more detailed account of the calculation leading to the O(1/n) form for lambda . They also examine the role of thermal fluctuations in the ordered phase and the effect of long-range initial correlations on the ordering process.