Inverse Laplace Transform of RLC Circuit
I'm no expert on this but I'd say you need to do more work on the denominator to make it fit a standard transform. Maybe: -

\$I(s) = \dfrac{6}{Ls^2+Rs+\dfrac{1}{C}}\$ becomes...

\$I(s) = \dfrac{\dfrac{6}{L}}{s^2+\dfrac{R}{L}s+\dfrac{1}{LC}}\$

Then convert denominator to \$(s + \dfrac{R}{2L})^2+(\dfrac{1}{LC} - \dfrac{R^2}{4L^2})\$

I'm not going further because I'm unsure - not done this stuff in ages

EDIT

Drawing attached to help understand the relationships between \$\omega_N\$ and \$\omega_0\$: -