Generalization of a Ramanujan identity

The Euler product for the Landau--Ramanujan constant could have motivated a curious identity by Ramanujan that appears in his notebooks two times. This observation involves a square root and the first four prime numbers of the form $4n+3$, i.e., $3, 7, 11, 19$. Berndt asks whether Ramanujan's identity is an isolated result, or if there are other identities of this type. With this work we would like to give a possible answer to Berndt's question.