Classical pixel-based Visual Servoing (VS) approaches offer high accuracy but suffer from a limited convergence area due to optimization nonlinearity. Modern deep learning-based VS methods overcome traditional vision issues but lack scalability, requiring training on limited scenes. This paper proposes a hybrid VS strategy utilizing Deep Reinforcement Learning (DRL) and optimal control to enhance both convergence area and scalability. The DRL component of our approach separately handles representation and policy learning to enhance scalability, generalizability, learning efficiency and ease domain adaptation. Moreover, the optimal control part ensures high end-point accuracy. Our method showcases remarkable achievements in terms of high convergence rates and minimal end-positioning errors using a 7-DOF manipulator. Importantly, it exhibits scalability across more than 1000 distinct scenes. Furthermore, we demonstrate its capacity for generalization to previously unseen datasets. Lastly, we illustrate the real-world applicability of our approach, highlighting its adaptability through single-shot domain transfer learning in environments with noise and occlusions. Real-robot experiments can be found at \url{https://sites.google.com/view/vsls}.