For $t=i$ there is a formula involving and 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 "Some theorems on stable processes" by Blumenthal and Getoor. Also see this related MO question.
For $t=i$ there is a formula involving and 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 "Some theorems on stable processes" by Blumenthal and Getoor. Also see this related MO question.