Any firearm made prior to 1969 did not require a serial number by law.
You will have to contact Savage directly for the answer to your question.I beleive that the serial numbers to the Stevens model 311 side by side shotgun are not in the public domain.I can say that the model 311 was discontinued in the year 1989.
Series H
1994
If it was manufactured after Dec 31, 1968, the serial number will be plainly marked on the receiver. There will be some numbers under the barrels, but they may serial numbers or just batch/lot numbers that will not help identify the date of manufacture. Serial numbers were not required on long arms in the US until 1969.
$250-$350.
I purchased my model 311 back in 1968, as a used shotgun. I was told it was factory made with no serial number. I dont know mfg date, but this will maybe start you in the back dating process.
There is no website to determine the age of a Stevens 311 by serial number. However, since yours has a serial number it is almost certainly post-1968. sales@countrygunsmith.net
100-350 US Dollars
The model designation was changed to 311 about 1941. Some were marked with both model numbers during the change.
The model 311 was made from 1941 to 1989, but since you have a serial number I would guess yours was made after the Gun Control Act of 1968. You could probably find out by contacting Savage Arms at the related link posted on the right side of this page.
Online
there is the Stevens side by side shotgun model 311