  By considering Schwarz's map for the hypergeometric differential equation
with parameters $(a,b,c)=(1/6,1/2,1)$ or $(1/12,5/12,1)$, we give some
analogies of Jacobi's formula $\vartheta_{00}(\tau)^2=
F(1/2,1/2,1;\lambda(\tau))$, where $\vartheta_{00}(\tau)$ and $\lambda(\tau)$
are the theta constant and the lambda function defined on the upper-half plane,
and $F(a,b,c;z)$ is the hypergeometric series defined on the unit disk. As
applications of our formulas, we give several functional equations for
$F(a,b,c;z)$.
