No mintmark, but minted in New Orleans (O) this was the first year with the mintmark on the reverse, and it's sometimes called 1840 w reverse of 38. A lovely, somewhat rare half dollar.
Me neither, and I posed the same question when I owned that piece. If I remember correctly, the Louisiana state sovereignty over the New Orleans Mint was the longest duration of those three separate periods within 1861, so based on probability alone, I guess it’s most likely to have been a Louisiana issue. I think there are some die markers and stuff that people have analyzed, but I don’t know anything much about all that stuff, or whether the diagnostics would even show on a relatively well circulated piece like the example above.