the lithium-cobalt composite oxide being represented by the formula Li_pCO_xM_yO_zF_a (wherein M is a transition metal element other than Co, aluminum or an alkaline earth metal element, 0.9≤p≤1.1, 0.980≤x≤1.000, 0≤y≤0.02, 1.9≤z≤2.1, x+y=1 and 0≤a=0.02),

which comprises firing a mixture of a cobalt source, a lithium source, and, if required, the above M element source and a fluorine source, at a temperature of from 800° C. to 1050° C. in an oxygen containing atmosphere,

characterized in that

the cobalt source comprises a first cobalt hydroxide and a cobalt oxyhydroxide, where the first cobalt hydroxide has a specific surface area of from 5 to 50 m²/g, a press density of from 1.0 to 2.5 g/cm³ and a structure having secondary particles formed by agglomeration of primary particles, and

further that the average particle size D50 after the secondary particles are dispersed in pure water is at most ¼ of the original average particle size,

wherein, as measured by means of X-ray diffraction in which CuKα is used as a radiation source,

of the first cobalt hydroxide, the half value width of a diffraction peak on (001) surface at 2θ=19±1° is from 0.18 to 0.35°, and the half value width of a diffraction peak on (101) surface at 2θ=38±1° is from 0.15 to 0.35°, and

of the cobalt oxyhydroxide, the half value width of the diffraction peak on the (220) surface at 2θ=31±1° is at least 0.8°, and the half value width of the diffraction peak on the (311) surface at 2θ=37±1° is at least 0.8°, and the specific surface area is from 10 to 80 m²/g.