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