Vacuum in algebraic quantum field theory

In algebraic quantum field theory (AQFT), the vacuum $\psi_0$ is the unique element of the Hilbert space that is Lorentz invariant and translation invariant. As a valid quantum state it is normalized $\|\psi_0\|=1$. The space of square-integrable functions $L^2$ is often used as a good representation of the Hilbert space. So, the vacuum should be one of these square-integrable functions. My question is, which one? Can somebody perhaps provide me with the expression of the function in $L^2$ that represents the vacuum that would illustrate its invariance and normalization?

I know that the vacuum can be represented in terms of a Wigner function as a Gaussian function on phase space, but that is not what I'm looking for because in that case operators are also Wigner functions, so we don't get the usual setting of AQFT. Moreover, a simple (one-dimensional) representation of phase space does not demonstrate the Lorentz invariance and translation invariance.

There has been several questions about the AQFT vacuum but I couldn't find anyone that addressed this question.