Hamiltonian from a Lagrangian with constraints?

Let's say I have the Lagrangian:$$L=T-V.$$Along with the constraint that $$f\equiv f(\vec q,t)=0.$$ We can then write:$$L'=T-V+\lambda f. $$What is my Hamiltonian now? Is it$$H'=\dot q_i p_i -L'~?$$Or something different? I have found at least one example where using the above formula gives a different answer then the Hamiltonian found by decreasing the degrees of freedom by one rather then using Lagrange multipliers.