I am looking for a function on the positive real line whose Laplace transform, with parameter $s$, is $$\left(\frac{\lambda}{1+\lambda}\right)^s,$$ where $s$ and $\lambda$ are greater than $0$.
The answer should be a probability density function on the positive real line, since the Laplace transform is $1$ when $s=0$. Further, when $s \rightarrow \infty$, the Laplace transform tends to $0$. So it is a Laplace transform.
Reference would also be appreciated. Thanks in advance.