You have:

L{cos(2 sqrt(t))/sqrt(pi t)} = exp(-1/s)/sqrt(s) ... (1)

Now we also know that:

LuF(ut) = f(s/u) ... (2)

Now let u=a and use (2) in (1):

L{a cos(2 sqrt(at))/sqrt(pi at)} = exp(-a/s)/sqrt(s/a)

............................................= sqrt(a) exp(-a/s)/sqrt(s).

Hence:

L{sqrt(a) cos(2 sqrt(at))/sqrt(pi at)} = exp(-a/s)/sqrt(s).

So the inverse LT of exp(-a/s)/sqrt(s) is sqrt(a) cos(2 sqrt(at))/sqrt(pi at).

RonL