You will need a professional appraisal.
Your browning sixteen gauge shotgun was made by FN for browning in 1949.
Gunbroker
the 5S prefix to your browning sweet sixteen shotgun puts the production of your shotgun in 1965.
Your browning 16ga auto-5 shotgun was made in 1934 with the serial number that you provided.However the browning sweet-sixteen auto-5 shotgun was not offered until 1937.These sweet-sixteen shotguns were made from 1937-1975.Does your browning shotgun say sweet-sixteen on the shotgun?If so please check your serial number again.
They made identical shotguns in other gauges, but if it is not a 16 gauge, it is an Auto-5, not a Sweet Sixteen.
Contact Browning Customer service and for @ 40 USD they can send you a historical letter.
The Browning Sweet 16 is a 16 gauge shotgun manufactured in Belgium. It uses a true 16 gauge barrel and as such is a wonderful shotgun. To a hunter this is a sweet weapon.
The serial number that you provided,indicates that you have a Browning auto-5 standard weight 16 gauge shotgun that was made in the year 1957.
100-400 USD depending on condition.
1959
The prefix of 2R was assigned to the Browning standard model 16 gauge auto-5 shotgun which was made in the year 1962.If you had a sweet sixteen model auto-5 shotgun,then the prefix to your serial number would have been 2S.Therefore you do not have a sweet-sixteen model Browning shotgun.
you would need to include if the shotgun in question is a sweet-sixteen auto-5 shotgun,or a browning side by side,or a standard weight auto-5 shotgun and the condition of your shotgun to get a accurate value .