Jaguar,
I've sent you a personal message. Please respond to it. There are NO LISTINGS for Spencer serial #s for the M1865 Spencer Models. Serial #s for the Spencer 1860 models and any firearm used during the Civil War can be found in the Springfield Research Service publication of SERIAL NUMBERS of US MARTIAL ARMS . There are four volumes 1, 2, 3, & 4. They have become very expensive to purchase, or even find.
Frank, I checked on your serial # 34048, two possible choices.
Your Spencer Carbine may have been issued to Company G, 18th New York Volunteer Cavalry on May 10, 1865, or it may have been issued to Company M, 2nd New York Volunteer Cavalry in 1865 (no day or month listed).
Please read my Spencer Serial # Disclaimer that is posted on the first page of my Spencer Shooting Society.
Two Flints