First eigenfunction on cylinder with mixed boundary conditions

Question

Consider $-\Delta u= \lambda_1 u$ to be a solution to the first eigenfunction on the cylinder $D^2\times [0,L]$ with mixed homogeneous boundary conditions. By mixed conditions, I mean Dirichlet on $D^2\times {0}\cup D^2\times {L}$, and Neumann otherwise. Is the first eigenfunction explicitly known in this case? I know that one can use separation of variables to directly compute in the Dirichlet case.