Without a calibrated monitor, you can not be sure of rendering accurate color. Even if you calibrate your monitor, only your monitor is guaranteed to be accurate and no other where the JPEG might be viewed. Saving the JPEG to the sRGB color space is the best you can do.
I would try to shoot the image outside on a cloudy day where there are no harsh shadows and artificial light is not necessary. It appears that your camera is perpendicular to the subject and so is your lighting. I would try a shot with three soft boxes positioned at equal angles from the coins AND closer to being in the same plane as the coins and not perpendicular to the coin face. This would allow the luster to be reflected from the coin back to the camera. If I can figure out how to represent this, I'll try to draw a lighting diagram. The lighting needs to have a color temperature of at least 5500˚K. CFLs are probably best at this.
Do you have an explanation as to why the obverse and the reverse of the two coins pictured have such a different cast?