Unless I am making a typo, the result is:
$$ 2^{2s} \frac{\Gamma(\tfrac n2+s) \Gamma(2s)}{\Gamma(\tfrac n2)^2} {_2F_1}(\tfrac n2+s, 2s, \tfrac n2, -|x|^2), $$
where $n$ is the dimension and ${_2F_1}$ is the Gauss's hypergeometric function. See Table 1 on page 168 in my survey [1], or Corollary 2 in the original paper [2].

References:

[1] Mateusz Kwaśnicki, *Fractional Laplace Operator and its Properties*, in: A. Kochubei, Y. Luchko, *Handbook of Fractional Calculus with Applications. Volume 1: Basic Theory*, De Gruyter Reference, De Gruyter, Berlin, 2019, https://doi.org/10.1515/9783110571622-007

[2] Bartłomiej Dyda, Alexey Kuznetsov, Mateusz Kwaśnicki, *Fractional Laplace operator and Meijer G-function*, Constructive Approx. 45(3) (2017): 427–448, https://doi.org/10.1007/s00365-016-9336-4