I've looked at several resources and used Wolfram alpha but have been unable to find a Laplace transform for the following function:
$$f(s) = {s\over \sqrt{a^2-\left({s\over 2}\right)^2}}$$
For a = 1 wolfram alpha found that
$$\frac{1}{4} (\pi \pmb{L}_{-1}(s)+\pi \pmb{L}_1(s)-2 \pi I_1(s)-4 i K_1(s)+2)$$
Where L is the modified struve function, I the modified Bessel function of the first kind and K the modified Bessel function of the second kind. Does this function have a Laplace function for general $a$?
Assuming $a>0$ and changing from $s$ to $x$ Mathematica gave a Meijer G function which FunctionExpanded down to $$ 2 i \pi a \left(\frac{1}{x^4}\right)^{3/4} x^3 \left(\pmb{H}_{-1}\left(\frac{2 a}{\sqrt[4]{\frac{1}{x^4}}}\right)+Y_1\left(\frac{2 a}{\sqrt[4]{\frac{1}{x^4}}}\right)\right) $$ with Struve H and Bessel Y functions. For $x>0$ you get $$ 2 i \pi a \left(\pmb{H}_{-1}\left(2ax\right)+Y_1\left(2ax\right)\right) $$ hope this helps.