I am testing some numerical algorithms for computing the Laplace-Beltrami eigenvalues on the sphere. One thing that came up was computing the first eigenvalue of the "equilateral" spherical triangle which realizes the $4$-tilling of the sphere. I could find some numerical computations giving the value $5.189..$. I found values close to this with my numerical method. In order to test my accuracy I wanted to find a more precise approximation of this eigenvalue.

I am interesting to know if an analytical formula for the first eigenvalue of such a triangle can be found.

Here is a picture representing one such triangle:

The slide is taken from this presentation: http://www.math.utah.edu/~treiberg/EigenvalCapture.pdf

The article in which the corresponding numerical computation is done can be found here: http://www.math.utah.edu/~treiberg/drunker-submit.pdf