Just like the answer of Tyberius, I suggest looking at the basis set data in the format of a popular software, instead of the general JSON format which is what is in the link you gave us. While the GAUSSIAN format is more compact, I think it's even more clear in the CFOUR format, which is the same way the data is presented if you choose MOLCAS, AcesII, DALTON, DIRAC, deMon2K, TURBOMOLE, MOLPRO, and some other places:
C:STO-3G
STO-3G Minimal Basis (3 functions/AO)
2
0 1
2 1
6 3
0.2941249355D+01 0.6834830964D+00 0.2222899159D+00 0.7161683735D+02 0.1304509632D+02 0.3530512160D+01
-0.9996722919D-01 0.0
0.3995128261D+00 0.0
0.7001154689D+00 0.0
0.0 0.1543289673D+00
0.0 0.5353281423D+00
0.0 0.4446345422D+00
0.2941249355D+01 0.6834830964D+00 0.2222899159D+00
0.1559162750D+00
0.6076837186D+00
0.3919573931D+00
Let me now explain what everything means:
2 # Number of types of functions (here we have S and P)
0 1 # Types of functions (0 = S-type, 1 = P-type)
2 1 # Number of contractions (2 S-type, 1 P-type)
6 3 # Number of primitives (6 S-type, 3 P-type)
Then we have the 6 S-type primitives followed by 12 S-type contraction coefficients, but notice that 6 of the contraction coefficients are 0, so that we are left with only 3 contraction coefficients for 1s and 3 contraction coefficients for 2s. This is why it's called STO-3G: There's 3 primitives for each orbital.
So the 1s orbitals are:
\begin{align}
\phi_{1s} &= c_{11} g_s(\alpha_1) + c_{21} g_s(\alpha_2) + c_{31} g_s(\alpha_3) + \color{gray}{c_{41} g_s(\alpha_4)+c_{51} g_s(\alpha_5)+c_{61} g_s(\alpha_6) }\\
\phi_{2s} &= \color{gray}{c_{12} g_s(\alpha_1) + c_{22} g_s(\alpha_2) + c_{32} g_s(\alpha_3)} + c_{42} g_s(\alpha_4)+c_{52} g_s(\alpha_5)+c_{62} g_s(\alpha_6),
\end{align}
where $\alpha_1,\alpha_2,\alpha_3,\alpha_4,\alpha_5,\alpha_6$ are given in this line:
0.2941249355D+01 0.6834830964D+00 0.2222899159D+00 0.7161683735D+02 0.1304509632D+02 0.3530512160D+01
Then the $c_{ij}$ matrix of coefficients is:
-0.9996722919D-01 0.0
0.3995128261D+00 0.0
0.7001154689D+00 0.0
0.0 0.1543289673D+00
0.0 0.5353281423D+00
0.0 0.4446345422D+00
Because of the 0.0
entries, we actually only have 3 terms for 1s and 3 terms for 2s, which is why it's called STO-3G.
So now your question was about the P-type orbitals. The 3 exponents ($\alpha$ in your question) are:
0.2941249355D+01 0.6834830964D+00 0.2222899159D+00
and the 3 contraction coefficients are:
0.1559162750D+00
0.6076837186D+00
0.3919573931D+00
and $\phi_{2p}$ is a sum of three terms.