Finding the value of a function at the global minimum

Question

I have the following function:

$f(y) = \frac{y^2}{2} - \frac{1}{a} * \ln(2\cosh(a*y))$

with $a>0$.

I want to find an analytic solution for the value of $f(y)$ at the global minimum $y_0$, namely $f(y_0)$.

For $a=1$ the numeric value would be $f(y_0) = -\ln(2)$ and for $a \rightarrow\infty$ it would be $f(y_0) = -0.5$.

For the position of the extrema I get:

$y_0 = \tanh(a*y_0)$

which - I think - cannot be solved analytically for $y_0$.

At the end, I want to get a function which gives me the value of the function $f(y)$ at the global minimum $y_0$ as a function of $a$.