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I know of only 4 official Swiss language: German, French, Italian, Romansch. Which one is the fifth? According to Wikipedia, German is the native language of its residents.
Swiss German is not a language, it's a dialect. However, in any case - you can count either German or Swiss German as one of the official languages, not both.
And yes, in Luzern the locals speak (Swiss) German.
There are slight differences between good German and the dialect of Lucerne.
Example: "Ich bin sehr gerne in Luzern geblieben" means in dialect "I by rüüdig gärn z'Lozärn blube".