Given that $$\mathcal{L} \lbrace J_0(t) \rbrace = \frac{1}{\sqrt{s^2+1}}$$ where $J_0(t) = \sum_{n=0}^{\infty}\frac{(-1)^n}{(n!)^2}\left(\frac{t}{2}\right)^{2n} \hspace{0.1in}$, find the Laplace transform of $\hspace{0.05in} tJ_0(t)$.
$\mathcal{L} \lbrace tJ_0(t) \rbrace =$ .