= Solution
At fixed volume, <entropy from heat capacity> obeys $dS=C(T)dT/T$. The <Third law of thermodynamics> fixes $S(0)=0$ when the ground state is nondegenerate, so below the transition
$$
S(T_c^-)=\int_0^{T_c}\frac{\alpha T}{T}\,dT=\alpha T_c.
$$
Above the transition,
$$
S(T)=S(T_c^+)+\int_{T_c}^{T}\frac\beta{T'}\,dT'
=S(T_c^+)+\beta\log\frac{T}{T_c}.
$$
For a transition with no <latent heat>, entropy is continuous, and comparison with $S=\beta\log T+\gamma$ gives the directly testable relation
$$
\boxed{\gamma=\alpha T_c-\beta\log T_c.}
$$
Thus calorimetric measurements of $C(T)$ and $T_c$ verify $\gamma$. More generally, if a measurable latent heat $L$ is present, $S(T_c^+)-S(T_c^-)=L/T_c$, and the formula becomes
$$
\boxed{\gamma=\alpha T_c+\frac{L}{T_c}-\beta\log T_c.}
$$
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