Interestingly neither of the chord finder apps quoted can give a name to the first chord even if you put an F on the bottom instead of an E.

I use this chord quite often in a spanish music context.

Also in jazz a maj7#11 chord is not uncommon so I don't know why it's not in the database.

If you just play the top four strings of this chord you get Fmaj7b5 (this is in the database)