Weibull function in Excel

I have a survival dataset (see below). I want to fit to Weibull function
S(t)=EXP(-alpha*Time^gamma). If I use Excel Solver, alpha=0.00367 and
gamma=2.32. If I use a method tranforming probability to Ln(Ln(1/S(t)) which
will be a linear function of Ln (t). But I got a different alpha and gamma.
When I compared two alphas adn gammas, the Solver results had a better
goodness of fit. What is the problem? THanks

Month Probability of survival
0 1
1 1
2 0.92
3 0.90
4 0.88
5 0.83
6 0.76
7 0.74
8 0.69
9 0.57
10 0.48
11 0.39
12 0.27
13 0.27
14 0.13