It only bears Smith and Wesson on left side of bbl, and 38 S and W special CTG. on right side.
Sounds like you don't have a S&W. You have something with a S&W barrel on it.
The usual places for S&W serial numbers are either the bottom of the grip frame or under the cylinder yoke.
I am unaware of anything S&W ever made for sale that didn't have a serial number, AND the S&W name on it.
There are only limited possibilities, the gun was made by S&W, or it wasn't.
There are guns that look like S&W but aren't.
Second, and more important, the gun either had markings that have been removed, including a serial number, or it never did.
And this is where it becomes a legal matter. Since 1968, every gun sold in the US has been required by law to have a serial number and the maker's name and address. The exception is, guns made before 1968 are not required to have a serial number UNLESS the maker put one on it.
A LOT of budget grade .22s and shotguns made before 1968 never had serial numbers, and are legal to own today without serial numbers.
BUT, a gun that had a serial number when made (no matter when made) must have a serial number on it now, or its not legal to own.
it is illegal to possess a firearm that has had the serial number "altered, defaced, or removed". No matter who did it, or when. If you have such a gun in your possession, YOU are breaking the law.
Doesn't matter if the serial number was "altered, defaced, or removed" in 1898 or last week, if you possess such a gun, you're breaking the law.
IF you find you are in possession of an illegal gun, the only legal course of action is to surrender the gun to law enforcement. DO NOT DO ANYTHING ELSE. DO NOT hammer it flat, melt it down, or throw it in the lake, no matter what anyone might tell you don't just "get rid of it" because doing so might be ANOTHER crime, as it could be destruction of evidence...
Generally speaking showing "good faith" in surrendering the gun (once you know, or suspect it is illegal to posses) rarely results in prosecution.
Its going to cost $ but I would recommend contacting a firearms lawyer, and following their advice.
You might have some rare valuable "factory goof". but I doubt it. You might have some handbuilt copy of a S&W frame fitted with S&W barrel. Unlikely but not totally impossible. You might have a S&W (or something else fitted with a S&W marked barrel) that was "scrubbed by the CIA" to remove all identifying markings.. unlikely as well, but even if it was, (and you can prove it) it's STILL NOT LEGAL for you to possess.
Check the gun carefully, inside and out. If you can't find a serial number, take it to the cop shop and give it to them, and answer any questions about where it came from, etc. You should be ok. Having a lawyer do that, on behalf of his client give you a bit more CYA, but costs. Your call, just remember the worst case is Felony conviction, prison time, fines and loss of firearm rights for life.
Good Luck!