The relation between the largest eigenvalue of $f's$ Hessian matrix and Lipschitz constant of the gradient of $f$

Question

Suppose that $f:\mathbb{R}^n\rightarrow\mathbb{R}$ is twice continuously differentiable, and the gradient of $f$ is Lipschitz continuous, i.e., \begin{align*} \|\nabla f(x)-\nabla f(y)\|_2\le L\|x-y\|_2, \forall\,x,y\in\mathbb{R}^n, \end{align*} where $L>0$ is the Lipschitz constant. Please find the relation between $L$ and the largest eigenvalue of $\nabla^2 f(x)$.