We present the second report of our systematic search for strongly lensed quasars from the data of the Sloan Digital Sky Survey (SDSS). From extensive follow-up observations of 136 candidate objects, we find 36 lenses in the full sample of 77,429 spectroscopically confirmed quasars in the SDSS Data Release 5. We then define a complete sample of 19 lenses, including 11 from our previous search in the SDSS Data Release 3, from the sample of 36,287 quasars with i<19.1 in the redshift range 0.6<z<2.2, where we require the lenses to have image separations of 1<theta<20 and i-band magnitude differences between the two images smaller than 1.25 mag. Among the 19 lensed quasars, 3 have quadruple-image configurations, while the remaining 16 show double images. This lens sample constrains the cosmological constant to be Omega_Lambda=0.84^{+0.06}_{-0.08}(stat.)^{+0.09}_{-0.07}(syst.) assuming a flat universe, which is in good agreement with other cosmological observations. We also report the discoveries of 7 binary quasars with separations ranging from 1.1 to 16.6, which are identified in the course of our lens survey. This study concludes the construction of our statistical lens sample in the full SDSS-I data set.
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