The Bolocam Galactic Plane Survey (BGPS) is a 1.1 mm continuum survey of dense clumps of dust throughout the Galaxy covering 170 square degrees. We present spectroscopic observations using the Heinrich Hertz Submillimeter Telescope of the dense gas tracers, HCO+ and N2H+ 3-2, for all 6194 sources in the Bolocam Galactic Plane Survey v1.0.1 catalog between 7.5 <= l <= 194 degrees. This is the largest targeted spectroscopic survey of dense molecular gas in the Milky Way to date. We find unique velocities for 3126 (50.5%) of the BGPS v1.0.1 sources observed. Strong N2H+ 3-2 emission (T_{mb} > 0.5 K) without HCO+ 3-2 emission does not occur in this catalog. We characterize the properties of the dense molecular gas emission toward the entire sample. HCO+ is very sub-thermally populated and the 3-2 transitions are optically thick toward most BGPS clumps. The median observed line width is 3.3 km/s consistent with supersonic turbulence within BGPS clumps. We find strong correlations between dense molecular gas integrated intensities and 1.1 mm peak flux and the gas kinetic temperature derived from previously published NH3 observations. These intensity correlations are driven by the sensitivity of the 3-2 transitions to excitation conditions rather than by variations in molecular column density or abundance. We identify a subset of 113 sources with stronger N2H+ than HCO+ integrated intensity, but we find no correlations between the N2H+ / HCO+ ratio and 1.1 mm continuum flux density, gas kinetic temperature, or line width. Self-absorbed profiles are rare (1.3%).
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