We consider the problem of private multiple linear computation (PMLC) over a replicated storage system with colluding and unresponsive constraints. In this scenario, the user wishes to privately compute P linear combinations of M files (each of length L) from a set of N replicated servers without revealing any information about the coefficients of these linear combinations to any T colluding servers, in the presence of S unresponsive servers that do not provide any information in response to user queries. Our focus is on a more practical case where the system parameters M, L, P are large, and hence the upload cost for user queries, as well as the computational complexities for servers and the user, are all significant and cannot be ignored. Unlike most previous literature that primarily focused on download cost from servers as a performance metric, we propose a novel PMLC scheme to establish a flexible tradeoff between communication costs and computational complexities.