We present an in-depth characterization of the polarimetric channel of the Large-Angle Spectrometric COronagraph/LASCO-C3 onboard SOHO. The polarimetric analysis of the white-light images makes use of polarized sequences composed of three images obtained through three polarizers oriented at +60$^circ$, 0$^circ$, and -60$^circ$, complemented by a neighboring unpolarized image. However, the degradation of the 0$^circ$ polarizer noticed in 1999 compelled us to reconstruct the corresponding images from the other ones thereafter. The analysis closely follows the method developed for LASCO-C2 (Lamy, et al. Solar Physics 295, 89, 2020 and arXiv:2001.05925) and implements the formalism of Mueller, albeit with additional difficulties notably the presence of a non-axially symmetric component of stray light. Critical corrections were derived from a SOHO roll sequence and from consistency criteria (e.g., the tangential direction of polarization). The quasi-uninterrupted photopolarimetric analysis of the outer corona over two complete Solar Cycles 23 and 24 was successfully achieved and our final results encompass the characterization of its polarization, of its polarized radiance, of the two-dimensional electron density, and of the K-corona. Comparison between the C3 and C2 results where their field of view overlaps shows an overall agreement. The C3 results are further in agreement with those of eclipses and radio ranging measurements to an elongation of about 10 solar radii but tend to diverge further out. Whereas the coronal polarization out to 20 solar radii is still highly correlated with the temporal variation of the total magnetic field, this divergence probably results from the increasing polarization of the F-corona.
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