A Model '97 with a serial number in the 354,000 range would have been shipped from the factory in 1906. What you see beside the .22 caliber marking is an S, L, & LR for short, long and long rifle cartridges that can be used in this rifle.
Value of any antique firearm is dictated by the amount and condition of the remaining original factory finish on the wood and metal. Without knowing these conditions, it is not possible to give an accurate estimation of value at this time. The best thing for you to do is go to some of the online internet gun auction sites like Auction Arms, Gun Broker and compare your guns condition to some of those listed and watch to see what they sell for. There is Guns America and Guns International that lists guns for sale too if you want to go there.
Due to the increasing cost of ammunition, there be no warning shot!
The growing federal deficit = generational slavery to the national debt.