I have a doubt about Laplace transforms. Most concretely, about the existence of an inverse.
I know already these facts
$$\mathcal{L}^{-1}(\exp(-\sqrt{s}))=\frac{\exp({-\frac{1}{4 t}})}{2 \sqrt{\pi } t^{3/2}}.$$ $$\mathcal{L}(\sin(-\sqrt{s})) \text{ does not exist}.$$
I was wondering if $$\mathcal{L}^{-1}(\exp(-\sqrt{s}) \cdot \sin(-\sqrt{s}))$$ exists or not. I know that if $\mathcal{L}(\sin(-\sqrt{s}))$ existed, then I could apply Convolution Theorem to get the result. But this is not the case, so I am not even sure about the existence.
Thank you in advance.