finding the computable function in s-m-n theorem.

The short answer is yes: $s_n^m$ itself is a computable function! That's the real content of the theorem. So you can just "run it" to calculate the value. Any (reasonable) proof of the s-m-n theorem should show you a way to compute it.

What the value is exactly is not too important however: it depends on the way you arithmetize computable functions (how you code computable functions for your "universal machine" $\phi^n_e(\dots)$).

(I would have posted this as a comment, but I don't have the reputation for it. If the above is not clear, let us know what source you are using. Then we can help you figure out what it should be for your particular formalization of computable functions.)

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