We searched high resolution spectra of 5600 nearby stars for emission lines that are both inconsistent with a natural origin and unresolved spatially, as would be expected from extraterrestrial optical lasers.The spectra were obtained with the Keck 10-meter telescope, including light coming from within 0.5 arcsec of the star, corresponding typically to within a few to tens of au of the star, and covering nearly the entire visible wavelength range from 3640 to 7890 Å. We establish detection thresholds by injecting synthetic laser emission lines into our spectra and blindly analyzing them for detections. We compute flux density detection thresholds for all wavelengths and spectral types sampled. Our detection thresholds for the power of the lasers themselves range from 3 kW to 13 MW, independent of distance to the star but dependent on the competing "glare" of the spectral energy distribution of the star and on the wavelength of the laser light, launched from a benchmark, diffraction-limited 10-meter class telescope. We found no such laser emission coming from the planetary region around any of the 5600 stars. As they contain roughly 2000 lukewarm, Earth-size planets, we rule out models of the Milky Way in which over 0.1% of warm, Earth-size planets harbor technological civilizations that, intentionally or not, are beaming optical lasers toward us. A next generation spectroscopic laser search will be done by the Breakthrough Listen initiative, targeting more stars, especially stellar types overlooked here including spectral types O, B, A, early F, late M, and brown dwarfs, and astrophysical exotica.