Is it known whether $e^{\frac{\pi^2}{12 \log 2}}$ is transcendental or algebraic?
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Sign up to join this communityIs it known whether $e^{\frac{\pi^2}{12 \log 2}}$ is transcendental or algebraic?
This number showed up in this other question.
This is most likely open, since alredy $e^{\pi^2}$ is not known to be transcendental.
As an added difficulty, I don't think that $\frac{\pi^2}{12 \log 2}$ is known to be transcendental either.
There are very few, very limited, tricks to prove this kind of result: things like taking $(-1)^{-i}$ and $i^i$ and applying Gelfond–Schneider, or building the Weierstrass $\wp$-function of $\mathbb{Q}(\sqrt{-d})$ to get the transcendence of $e^{\pi\sqrt{d}}$ from its invariants.