This should narrow it down to two choices as Franklin E. is too old.
Franklin Jones, 18; Oct. 6, 1862 pvt., Co. F, 124th Ohio Infantry Rgt. July 18, 1863 discharged.
Franklin Jones, 19; Feb. 1, 1864 pvt., Co. A, 122nd Ohio Infantry Rgt. Dec. 10, 1864 discharged for disability.

Franklin E. Jones, 39; May 2, 1864 pvt., Co. H, 159th Ohio Infantry Rgt. Aug. 22, 1864 mustered out.

Source: Official Roster of the Soldiers of the State of Ohio