Colt made an 1878 model and an 1883 model.
The 1883 was a hammerless gun.
All these Colt shotguns had serial numbers like almost all other Colt firearms.
If the serial number actually is number 3, it would have value to a collector.
Check the gun over carefully to see if there's another serial number on it.
The Model 1878 hammer gun was available in 10 or 12 gauge with barrels from 28 to 32 inches.
The barrels were browned or blued damascus steel.
It had double triggers, side plates, and a color case hardened receiver.
The stock was a semi-pistol grip style.
Colt made 22,683 of these guns between 1878 and 1889.
Many of these guns were custom order models with custom options, and the custom order guns are worth more money.
In order to identify whether yours is a custom order gun, you'd have to buy a Colt Archive letter. Colt will do a hand search of the Colt records for your specific gun and send you a letter detailing everything Colt knows about the gun, including in what configuration if was shipped, whether it had custom features, when it was shipped, and who it was shipped too.
An Archive letter for a shotgun costs $100.00
http://www.coltsmfg.com/CustomerServices/ArchiveServices.aspx
Values for the standard production Model 1878 are based on the percentage of remaining original finish, and on the gun being undamaged.
Values run from a low of a 10% gun at $725, to over $4,000 for one in 100%.
A custom order gun would be more.
Note that these guns were made with damascus barrels and are NOT safe to shoot with any modern shells, and should not be fired with black powder shells unless and until a shotgun gunsmith who's expert with damascus guns says it's safe. This usually requires x-rays of the barrels.
No matter how good the gun may look or how nice the condition of the barrels seems, these barrels can have hidden flaws inside the damascus steel laminations that can allow them to explode.
These guns were made for use with black powder and modern shells use powders that have a very different pressure curve that can easily blow them up.