No one law encompasses all of the towns in that area. Much of it is shotgun only, so a muzzle loader works. Handgun calibers are specific to the 357, 41, and 44 magnums, and the 45 Colt. These calibers are okay in most-but not all-shotgun only turf. Your 500 is limited to regular rifle season in areas that are not shotgun only.
You can find more info on NH Fish & Game's website.