Whoa now folks. Seems to be some misconceptions floating around here. 1935F is the series designation, nothing more. The year (1935 on your note) used to not be changed unless there was a design change of the note itself. A new letter was added each time a Treasurer or a Secretary of the Treasury was changed. The 'year' 1935 ran from 1935 until 1957, when the $1 silver certificates were issued with series 1957, even though there was no design change. 1935H notes were printed up until 1957 as new notes. However in more recent times, the BEP basically abandoned that method and will change the year when a new Secretary of the Treasury is appointed, but add a letter each time only the Treasurer changes. (also it gets a bit more complicated when you look into the new colorized series of the larger ($5 and up) denominations. So I will use the $1 notes as an example.
Example: ($1 FRN notes)
series 1993: Withrow-Bentsen signatures.
series 1995: Withrow-Rubin signatures. (new series because the Secretary of the Treasury is changed but not the Treasurer)
series 1999: Withrow-Summers signatures. (again, Secretary changes, so new series year)
series 2001: Marin-Oneill signatures. (again, new Secretary, so new series year)
series 2003: Marin-Snow signatures. (once again, new Secretary, new series year)
series 2003A: Cabral-Snow signatures. (here, a new Treasurer is appointed but not a new secretary, so we get a 2003A series)
series 2006: Cabral-Paulson signatures. (currently being printed, new secretary, new series).
Lastly I should mention that the silver certificates are NOT federal reserve notes, so therefore have no issuing federal reserve bank.
The 12 FRBs that are current are the same 12 FRBs that were originally designated in 1913 when the Federal Reserve Banking System was adopted.