Designation is wrong, that is not a "standard series" size. It will be called a 3.625-16 UN, not UNC. You may need to call and have a custom tap made, or thread mill the part.
BT is correct about the designation. By the way, it is NOT a special, either, it is standard designation. Quickest way to cut that thread would be to single-tool it on a cnc lathe, if it is a checkable and turnable part.
As others have said, single-pointing (lathe) or thread milling (machining center) would be better approaches. If lot size was small enough, you could even threadmill with the single-point tool (slow).
The tap would be an expensive, single-purpose tool.
Power House Tools has 1"-7" there 8 threads inch but you might can get them to make 3 5/8-16 if you need. My boss has 1-6 1/2 in them quite price from what I understand.
Depending on your budget a collapsible tap would work for this job if you cannot single point. Its been years since I have seen one in use but they are still available. You may even find one on the used market and have the inserts supplied to you.