On the $L_{\mathrm{Y}\mathrm{J}}(\xi ,\eta ,X)$ constant for the Banaś-Frączek space

In this paper, for any $\lambda \ge 1,R_{\lambda }^2$ is the Banaś-Frączek space. The exact value of $L_{\mathrm{Y}\mathrm{J}}(\xi ,\eta ,X)$ for this space will be calculated. Specifically, $L_{\mathrm{Y}\mathrm{J}}(\xi ,\eta ,R_{\lambda }^2)=1+\frac{2\xi \eta }{{\xi }^2+{\eta }^2}(1-\frac{1}{{\lambda }^2})$ is the result thereafter through meticilous computation.