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We consider the free energy $F(\beta)$ of the directed polymers in randomenvironment in $1+1$-dimension. It is known that $F(\beta)$ is of order$-\beta^4$ as $\beta\to 0$. In this paper, we will prove that under a certaincondition of the potential, \begin{align*} \lim_{\beta\to0}\frac{F(\beta)}{\beta^4}=\lim_{T\to\infty}\frac{1}{T}P_{\mathcal{Z}}\left[\log\mathcal{Z}_{\sqrt{2}}(T)\right] =-\frac{1}{6}, \end{align*} where$\{\mathcal{Z}_\beta(t,x):t\geq 0,x\in\mathbb{R}\}$ is the unique mild solutionto the stochastic heat equation \begin{align*} \frac{\partial}{\partialt}\mathcal{Z}=\frac{1}{2}\Delta \mathcal{Z}+\beta \mathcal{Z}{\dot{\mathcalW}},\ \ \lim_{t\to 0}\mathcal{Z}(t,x)dx=\delta_{0}(dx), \end{align*} where$\mathcal{W}$ is a time-space white noise and \begin{align*}\mathcal{Z}_\beta(t)=\int_\mathbb{R}\mathcal{Z}_\beta(t,x)dx. \end{align*}

Free energy of directed polymers in random environment in $1+1$-dimension at high temperature