Navigating to a designated goal using visual information is a fundamental capability for intelligent robots. To address the practical demands of multi-modal, open-vocabulary goal queries and multi-goal visual navigation, we propose LagMemo, a navigation system that leverages a language 3D Gaussian Splatting memory. During a one-time exploration, LagMemo constructs a unified 3D language memory with robust spatial-semantic correlations. With incoming task goals, the system efficiently queries the memory, predicts candidate goal locations, and integrates a local perception-based verification mechanism to dynamically match and validate goals. For fair and rigorous evaluation, we curate GOAT-Core, a high-quality core split distilled from GOAT-Bench. Experimental results show that LagMemo's memory module enables effective multi-modal open-vocabulary localization, and significantly outperforms state-of-the-art methods in multi-goal visual navigation. Project page: https://weekgoodday.github.io/lagmemo