For $t=i$ there is a formula involving an integral of a Bessel function, so I doubt
there is a simple closed formula for $K_{\alpha}$ in general. You can find the formula I mentioned in the first page of the article <a href="http://www.ams.org/journals/tran/1960-095-02/S0002-9947-1960-0119247-6/">"Some theorems on stable processes"</a>
by Blumenthal and Getoor. Also see <a href="http://mathoverflow.net/questions/91263/asking-for-a-fourier-inverse-transform-which-is-related-to-stable-laws/91401#91401">this related MO question</a>.