The classification of $(g,K)$-modules tells you that the Beltrami-Laplace eigenvalue

$$\lambda = (1-s)s$$

of Maass form $f$ (of weight $0$) satsifies either $$\Re s = 1/2$$ or

$$0 < s <1,$$ that for the adelic decomposition
$$f = f _\infty \otimes \bigotimes_p f_p,$$
the function $f_\infty$ is is a vector of the induced representation
$$Ind_{B}^G | \cdotp |^{s-1/2}$$
at the infinite place. Of course, this is slight overkill and follows also from the positivity of the Beltrami Laplace operator.

However, it explains better the Selberg eigenvalue conjecture $0 \neq \lambda \geq 1/4$. The Selberg eigenvalue conjecture asserts that $0 < s <1$ or eq. $0 < \lambda \leq 1/4$ should imply $s=1/2$ or eq. $\lambda =1/4$,. The eigenvalue $\lambda = 1/4$ does in fact occur, Bump gives an example in chapter 1.

Selberg eigenvalue conjecture can be generalized also to the $p$-adic places, and is then called the Ramanujan-Petersson conjecture.

The difference between $\Re s = 1/2$ and $\Re s \neq 1/2$ is that the former is tempered and the latter is not.

There are some non trivial bounds towards the Selberg eigenvalues known. Functoriality of the $n$-th symmetric tensor would imply the result, and actually this is what Kim and Sarnak used to prove a non-trivial bound.

Furthermore the representation $Ind_{B}^G | \cdotp |^{s-1/2}$ are not square integrable, but the discrete series are. That is why the name discrete series, they occur discretely in the right regular representation of $L^2(G)$.