  The paper is concerned about a sharp form of Anisotropic Moser-Trudinger
inequality which involves $L^{n}$ norm. Let \begin{equation*}
\lambda_{1}(\Omega) = \inf_{u\in W_0^{1,n}(\Omega),u\not\equiv 0} ||F(\nabla
u)||_{L^n(\Omega)}^n / ||u||_{L^n(\Omega)}^n \end{equation*} be the first
eigenvalue associated with $n$-Finsler-Laplacian. using blowing up analysis, we
obtain that \begin{equation*} \sup_{u\in W_{0}^{1,n}(\Omega),||F(\nabla
u)||_{L^n(\Omega)} = 1} \int_{\Omega}e^{\lambda_n (1+\alpha||u||_{L^n
(\Omega)}^n)^{\frac{1}{n-1}} |u|^{\frac{n}{n-1}}}dx \end{equation*} is finite
for any $0\leq \alpha<\lambda_{1}(\Omega)$,and the supremum is infinite for any
$\alpha\geq \lambda_{1}(\Omega)$, where $\lambda_{n} = n^{\frac{n}{n-1}}
\kappa_n^{\frac{1}{n-1}}$ ($\kappa_{n}$ is the volume of the unit wulff ball)
and the function $F$ is positive,convex and homogeneous of degree $1$, and its
polar $F^o$ represents a Finsler metric on $\mathbb{R}^n$. Furthermore, the
supremum is attained for any $0\leq \alpha<\lambda_{1}(\Omega)$.
