The tendency of Large Language Models (LLMs) to generate hallucinations raises concerns regarding their reliability. Therefore, confidence estimations indicating the extent of trustworthiness of the generations become essential. However, current LLM confidence estimations in languages other than English remain underexplored. This paper addresses this gap by introducing a comprehensive investigation of Multilingual Confidence estimation (MlingConf) on LLMs, focusing on both language-agnostic (LA) and language-specific (LS) tasks to explore the performance and language dominance effects of multilingual confidence estimations on different tasks. The benchmark comprises four meticulously checked and human-evaluate high-quality multilingual datasets for LA tasks and one for the LS task tailored to specific social, cultural, and geographical contexts of a language. Our experiments reveal that on LA tasks English exhibits notable linguistic dominance in confidence estimations than other languages, while on LS tasks, using question-related language to prompt LLMs demonstrates better linguistic dominance in multilingual confidence estimations. The phenomena inspire a simple yet effective native-tone prompting strategy by employing language-specific prompts for LS tasks, effectively improving LLMs' reliability and accuracy on LS tasks.