We present results for QSO PKS 0405-123 (z=0.574, V=14.9), as part of a STIS Investigation Definition Team (IDT) key project to study weak Ly-alpha forest systems at low z. We detect 59 (47) Ly-alpha absorbers at 4.0 sigma significance to an 80% completeness limit of column density log N(HI)=13.3 (13.1) for Doppler parameter V_Dop=40 km/s over 0.002<z<0.423 (0.020<z<0.234). We find 4 intervening O VI systems, useful for studies of hot intergalactic gas. We do not distinguish between metal and Ly-alpha-only systems in the following analysis. The redshift density is consistent with previous measurements for log N(HI)>=14.0, but exhibits twice as many systems at 13.1<log N(HI)<14.0 compared to the mean number density of lines at z<0.07 toward 15 extragalactic objects. The difference possibly arises from cosmic variance. The Doppler parameter distribution has <V_Dop>=48 +- 21 km/s; line blending possibly inflates the value. We find evidence for Ly-alpha-Ly-alpha clustering in our sample on a scale of Delta v<=250 km/s, and there is evidence for a void at 0.032<z<0.081 with probability of occurrance P=0.0005. We find line-of-sight velocity correlations of up to 250 km/s between Ly-alpha absorbers with log N(HI)>=13.1 and 45 galaxies taken from the literature and unpublished data at 0<z<0.47; the transverse distances cover up to 1.5 /h_70 Mpc in the local frame. The Ly-alpha-galaxy clustering is stronger for higher log(N(HI) systems.
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