Abstrakt / Abstract: Context. The original 1/a-distribution is the only observational basis for the origin of long-period comets (LPCs) and the dynamical properties of the Oort Cloud. Although they are very subtle in the motion of these comets, non-gravitational effects can cause major changes in the original semimajor axis, 1/a_ori. Aims: We obtained reliable non-gravitational orbits for as many LPCs with small perihelion distances of q < 3.1 au as possible, and determined the corresponding shape of the Oort spike. Methods: We determined the osculating orbits of each comet using several data-processing methods, and selected the preferred orbit using a few specific criteria. The distribution of 1/a_ori for the whole comet sample was constructed using the individual Gaussian distribution we obtained for the preferred solution of each comet. Results: The derived distribution of 1/a_ori for almost all known small-perihelion Oort spike comets was based on 64% of the non-gravitational orbits. This was compared with the distribution based on purely gravitational orbits, as well as with 1/a_ori constructed earlier for LPCs with q > 3.1 au. We present a statistical analysis of the magnitudes of the non-gravitational acceleration for about 100 LPCs. Conclusions: The 1/a_ori-distribution, which is based mainly on the non-gravitational orbits of small-perihelion Oort spike comets, is shifted by about 10 × 10-6 au-1 to higher values of 1/a_ori compared with the distribution that is obtained when the non-gravitational effects on comet motion are ignored. We show the differences in the 1/a_ori-distributions between LPCs with q < 3.1 au and those with q > 3.1 au. These findings indicate the important role of non-gravitational acceleration in the motion and origin of LPCs and in the formation of the Oort Cloud. Full Tables 1, 3, and A.1-A.4 are only available at the CDS via anonymous ftp to http://cdsarc.u-strasbg.fr (ftp://130.79.128.5) or via http://cdsarc.u-strasbg.fr/viz-bin/cat/J/A+A/633/A80.