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We present STIS 7km/s resolution data of the Ly-a forest toward PKS0405-123(z=0.574). We make two samples. The strong sample (60 systems) has columndensity logNHI>13.3 over 0.002<z<0.423, and the weak sample (44) haslogNHI>13.1 at 0.020<z<0.234; 7 absorbers show metals, all with OVI, but oftenoffset in velocity from HI. The strong sample Doppler parameters show<b>=47+-22km/s; for the weak, <b>=44+-21km/s. Line blending and s/n effectslikely inflate b. The redshift density dN/dz is consistent with previous, lowerresolution results for logNHI>14.0. For 13.1<logNHI<14.0, we find a dN/dzoverdensity of ~0.2-0.3 dex at 0.127<z<0.234, which we believe arises fromcosmic variance. We find Ly-a clustering over Delta v<250km/s for logNHI>13.3,consistent with a numerical model. There is a void in the strong sample at0.0320<z<0.0814. We detect radial velocity correlations for Delta v<250km/sbetween Ly-a absorbers and 39 galaxies in a 10' field. The correlation growswith minimum NHI, peaking for logNHI ~> 13.5-14.0. The strength is similar tothat of the galaxy-galaxy correlation for our sample, implying that such Ly-asystems have mass log M/Msun= 11.3+1.0-0.6. Including lower column densitysystems in the sample shows correlations only out to Delta v <125km/s, as wouldbe expected for smaller density perturbations. Galaxy counts and local HIcolumn density appear correlated, peaking at ~5000km/s. We also give columndensities for Galactic species. (Abridged)Comment: 59 pages, 6 tables, 14 figures, 1 standalone, ApJ, in press; minor  revisions/clarifications for acceptance, including galaxy lis

The Low‐Redshift Lyα Forest toward PKS 0405−123