We present results of a mapping observation of the Galactic center region in the CO (J = 2-1) line at 230 GHz, using the Tokyo-NRO 60 cm survey telescope (9' beam). The CO (J = 2-1)/CO (J = 1-0) intensity ratio, R((2-1)/(1-0)), exceeds unity for the Galactic center molecular gas with bright CO emission. The intermediate-density [n(H-2) greater than or similar to 10(3) cm(-3)] molecular gas traced by CO (J = 2-1) line shows a highly asymmetric pattern in the I-V plane. This asymmetry appears more prominently in CS and NH3 line emissions which probe denser molecular gas. Nevertheless, sites of recent massive star formation traced by hydrogen recombination lines (H n regions) and evolved intermediate-mass stars traced by OH maser emission (OH/IR stars) show a relatively symmetric pattern in the I-V plane. This simple kinematics indicates that star formation has been taking place, continuously or intermittently, in a region about 100 pc in radius (''Star-forming ring''). A closer comparison shows that the H n regions have systematically lower velocities (in absolute values) than the OH/IR stars. If the kinematics of OH/IR stars reflect the galactic gravitational potential, this difference can be understood as a result of bisymmetric distribution of the H n regions. A possible interpretation is that the outer region of the star-forming ring is near the first inner Lindbrad resonance (ILR), where molecular gas accumulates in two arms to be self-gravitating and intense star formation is triggered, which we see as ''twin peaks'' in the central regions of barred galaxies. This small ILR radius requires the existence of a small-scale (similar to 1 kpc), rapidly rotating bar in the central region of the Galaxy. We calculated the intensity ratio between our CO (J = 2-1) data and the CO (J = 1-0) data taken with the Columbia 1.2 m telescope. The average ratio for the gas at -3 degrees less than or equal to l less than or equal to +3 degrees within 4.'5 (10 pc) from the Galactic plane is 0.74 +/- 0.02. Regions with R((2-1)/(1-0)) greater than or equal to 1.2 are found along the high-velocity side of the CO emission. The cloud complex located close to Sgr A* (Sgr A molecular cloud complex), which contains two well-known giant clouds (20 km s(-1) and 40 km s(-1)) exhibit an extremely high intensity ratio R((2-1)/(1-0)) congruent to 1.4. An anticorrelation is seen in the l-V diagram between the gas with high ratio [R((2-1)/(1-0)) greater than or equal to 1.2] and H II regions [R((2-1)/(1-0)) similar to 1]. The observed large R((2-1)/(1-0)) exceeding unity can be interpreted as different sizes of ''photospheres'' for the two lines in clumps which compose the Galactic center clouds.