From the Internet article, "Is my Lee Enfield sniper rifle a fake?"
... Caveat Emptor.
Canadian Long Branch rifles closely parallel British production, with some specific deviations and much smaller quantities. Rifles delivered before May 1944 were missing the characteristic ‘T’ stamp on the sidewall. In 1943 Long Branch delivered 71 sniper rifles with Canadian-made Research Enterprises Ltd. (REL) C No.32 MK.1 scopes, possibly in the 34L###x serial number range. These first deliveries were selected from the nearly 230,000 No.4 rifles produced in 1943 (numbered 23L###x to 56L###x). Law’s book suggests four distinct serial number ranges for No.4 Mk1*(T)'s: 1944 production rifles numbered 71L###x with REL C No.32 MK. 1 and 2 scopes; 350 odd 1944 production rifles numbered 74L0001 to 74L0350 with civilian production Lyman Alaskan scopes (bought as a stop gap when REL couldn't deliver fast enough); approximately 84 1944 production rifles numbered 80L8### with the REL C No.67 Mk.I scope; and a final batch of 376 1945 dated 90L8### rifles with REL C No.32 MK.3 scopes. "
A 1943 LB is plausible, but 37L is not within the recognized serial number range. Mistakes are possible in the references.