Why the t-test for a specific gene shows different value compared to differential analysis?
There is no reason your t-test should reproduce edgeR. In fact, edgeR exists because t-test is inappropriate.

edgeR does the tests by pooling information from all genes, because with the low number of replicates your t-test doesn't give sufficient statistical power.

What you need to do is: check visually your gene and make a decision yourself based on graphical results and edgeR results if your gene shows biologically significant.