Eigenvalues/vectors of the Laplace transform?

Question

I'm learning about eigenvalues and eigenvectors (finally starting to get them). This might be a silly question, but what is/are the eigenvector(s) of the Laplace transform? I mean, what $\vec{x}_{i}$'s and $\lambda_{i}$'s satisfy \begin{align}\mathcal{L}\left\{\vec{x}_{i}\right\}&=\lambda_{i}\vec{x}_{i}.\end{align} I'm just trying to extrapolate a bit from the fact that \begin{align}D_{t}e^{\lambda t}&=\lambda e^{\lambda t},\end{align} but I cannot think of any function that remains unchanged under the transformation.

Answer 1

The Laplace transform of $t^p$ is proportional to $\frac{1}{s^{p+1}}$ for $p>-1$. Take $p=-1/2$.