According to the definitive book on the subject titled L.C. Smith Shotguns by Lt. Col. William S. Brophy, USAR Ret.:
"The first L.C. Smith barrels were either Damascus or Twist steel. Neither of them is considered safe, by today's standards, for use with modern shotshells and should be retired from service.
"In descending order, these barrels were described as: Finest Damascus, Very Fine Damascus, Extra Fine Damascus, Fine Damascus, Good Damascus, and Best English Stub Twist.
"It is alleged that L.C. Smith barrels were frequently made to look like Damascus steel to please the sportsman who thought that Damascus was the best and strongest type of steel, or for the gun fancier who liked the swirl and curl pattern of Damascus steel. But this allegation is not true. NOTE: L.C. Smith shotgun barrels were never made with a phony patter which was put on the barrels to make them look like Damascus steel. If your barrels have the distinct pattern of Damascus or Twist steel, they are genuine!!"
Brophy's book has extensive model/serial number tables. If you can provide the gun's model and serial number, then I can look up the date of manufacture for you.
In the meantime, please take his admonition to heart and DO NOT attempt to shoot this gun using today's shotshells. The result could be a burst barrel which could cause injury or be fatal.
If your heirloom is a high-grade L.C. Smith, I recommend having it thoroughly vetted by a competent gunsmith. He/she can evaluate its safety for use with certain black powder loads, or the possibility of having the barrels sleeved so that it may be safely used with today's shotshells.
Good luck, and good shooting (but only with a safe gun)!