Are dimensionless constants special values of L-functions?

Speculative — an open question, not a coordinate. Surfaced in the first, undisciplined exploration; recorded here stripped of the overreach it arrived with.

The question. Is there a class of dimensionless constants that are literally special values of L-functions — motivic periods, in the Beilinson/Deninger picture, of which ζ(2) = π²/6 is the toy case — and can that class be characterized cleanly?

Separated from. The chat's version was entangled with a retracted overstatement (a claimed α–G₂ relation). That physics claim is dropped; the residue worth keeping is the purely arithmetic one — whether "dimensionless = special L-value" picks out a real class.

Discriminator (the gate). A genuine instance is a constant provably equal to an L-value or a period, with the theorem named; a numerical near-equality is not. Until a class is characterized — which constants, forced by what — this stays a question, not a claim.