Function equal to his Laplace or Mellin transform As the title suggests I am looking for functions $f(s)$ for which $\mathscr{L}\{f(s)\}=f(s)$ or $\mathcal{M}\{f(s)\}=f( s)$ where $\mathscr{L}$ and $\mathcal{M}$ stand respectively for Laplace transform and Mellin transform.

edit:
  I confirm it is an « or ». More precisely I am looking for non zero functions invariant for some Integral Transform. For Mellin or Laplace transform, if there is not such functions can you please provide a proof or a hint. Merci

 A: The function
$$f(s)=\frac{1}{\sqrt{t}}$$
has a Laplace transform 
$$F(t) = \frac{\sqrt{\pi}}{\sqrt{s}}$$ 
which is correct up to a constant.

Attempt 2: Note that
$$ \mathcal{L}(t^z) = s^{-z-1} \Gamma(l+1)$$ 
when $\Re (z)>1$. So if now $0<\Re (z)<1$, let
$$f(t)=\sqrt{\Gamma(z)} t^{-z} + \sqrt{\Gamma(1-z)} t^{z-1}$$
so that we get a Laplace transform
\begin{align}
F(s) &= \sqrt{ \Gamma(z)  \Gamma(1-z) } f(s)\\
&=\sqrt{ \frac{\pi}{\sin(\pi z)}} f(s)
\end{align}
So if 
\begin{align}
\frac{\pi}{\sin(\pi z)} &= 1\\
\implies z &= \frac{\arcsin(\pi)}{\pi}
\end{align}
We have a nontrivial function whose Laplace transform equals itself.
A: Edit - Not that it makes a difference to the solution I've presented, I will appease those commenters here by modifying my original solution to accomodate 'or' over 'and'... 

Not sure if you can do anything here in it's current definition
Here you have the following identity that you wish to find functions to comply with:
\begin{equation}
\mathscr{L}_{t \rightarrow s} \left\{ f(s) \right\}(s) = f(s) \lor \mathcal{M}_{t \rightarrow s} \left\{f(s) \right\}(s) = f(s) \nonumber 
\end{equation}
Given the linearity of both the Laplace and Mellin Transformations this becomes:
\begin{equation}
\mathscr{L}_{t \rightarrow s}  \left\{ 1\right\}(s) = 1  \lor \mathcal{M}_{t \rightarrow s} \left\{1 \right\}(s) = 1 \nonumber 
\end{equation}
Which is false as:
\begin{equation}
 \mathscr{L}_{t \rightarrow s}  \left\{ 1\right\}(s) = \frac{1}{s}\nonumber
\end{equation}
And the Mellin Transformation doesn't exist (outside of $f(s) = 0$)
As such, under your current definition there are no functions that can satisfy it. 
