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:
![enter image description here][1]

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

  [1]: https://i.sstatic.net/416Yr.png