For $n=3$, $$\lambda(r)=\frac{\pi^2}{r^2}-1,\ 0<r\leq\pi.$$

Unfortunately, the first Dirichlet eigenvalue for spherical caps in other dimensions cannot be calculated explicitly, but some estimates are known: $$\lambda(r)=\frac{j_{(n-2)/2,1}^2}{r^2}+O(1)\quad\text{as } r\to0^+,$$
where $j_{\nu,1}$ is the first zero of Bessel function $j_\nu$, and as $r\to\pi^{-}$,
$$\lambda(r)=\begin{cases}c_{n}\left(\pi-r\right)^{n-2}+o\left(\left(\pi-r\right)^{n-2}\right),\quad & n\geq3;\\ -c_{2}\log^{-1}(\pi-r)+o\left(\log(\pi-r)^{-1}\right),\quad &n=2.
\end{cases}$$
for some constants $c_n$. The proof and more sharper estimates on $\lambda(r)$ can be found in a [paper][1] by Borisov and Freitas.  


  [1]: https://dx.doi.org/10.4310/CAG.2017.v25.n3.a1