Assuming it’s a Model 1860 Spencer Carbine in .56-56 (as opposed to one of the later M1865 carbines made under Burnside’s contract in .56-50), the earliest date the carbine may have been delivered into Federal stores within this serial number range was in August 1864. While I cannot provide more info anent what military unit it might have been issued, if such a record exists, you may find it of interest that a M1860 Spencer carbine (SN 34617) was seized by Federal troops at Lloyd’s Tavern in April 1865 having belonged to Lincoln assasin John Wilkes Booth. Another Spencer carbine (SN 24304) was recovered after Booth's death, still gripped in his hand. Booth had purchased the two Spencer carbines during a trip to New York in January 1865 together with six Colt revolvers, and three Bowie knives, along with caps, cartridges and belts, and two pairs of handcuffs. As this demonstrates, some Spencer firearms were not issued to military forces but found their way instead into the waiting hands of criminals.

http://books.google.com/books?id=86RH6NNCl0QC&pg=PA187&lpg=PA187&dq=Lloyd%E2%80%99s+Tavern,+John+Wilkes+Booth+Spencer&source=bl&ots=j_LYREzIYY&sig=Ga3T7NclEDf6pqiVjJebmrF6t88&hl=en&ei=hXM4SpmXO4PBtwf8yYzeDA&sa=X&oi=book_result&ct=result&resnum=4