Neural codec language models, built on transformer architecture, have revolutionized text-to-speech (TTS) synthesis, excelling in voice cloning by treating it as a prefix continuation task. However, their limited context length hinders their effectiveness to short speech samples. As a result, the voice cloning ability is restricted to a limited coverage and diversity of the speaker's prosody and style. Besides, adapting prosody, accent, or appropriate emotion from a short prefix remains a challenging task. Finally, the quadratic complexity of self-attention limits inference throughput. In this work, we introduce Lina-Speech, a TTS model with Gated Linear Attention (GLA) to replace standard self-attention as a principled backbone, improving inference throughput while matching state-of-the-art performance. Leveraging the stateful property of recurrent architecture, we introduce an Initial-State Tuning (IST) strategy that unlocks the possibility of multiple speech sample conditioning of arbitrary numbers and lengths and provides a comprehensive and efficient strategy for voice cloning and out-of-domain speaking style and emotion adaptation. We demonstrate the effectiveness of this approach for controlling fine-grained characteristics such as prosody and emotion. Code, checkpoints, and demo are freely available: https://github.com/theodorblackbird/lina-speech