So... is Servant x Service finished or not?
NO, it is not finished.
Why?
The anime finished before the manga:
Anime finished: September 28, 2013
Manga finished: June 25, 2014
MangaUpdates contains information about the anime adaptation and indicates that it is incomplete.
Watching Order (by release date)
Here you have all the anime relations we detected ordered by release date, so you can see the reason for this status in context and also use it as watching order.
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Synopsis
Servant x Service is a workplace comedy focusing on three new employees at a Japanese government office. These newcomers each have their own unique reasons for joining the civil service.
Lucy is driven by a personal vendetta, seeking revenge against a civil servant from her past. Yutaka is more interested in finding ways to avoid work, while Saya is genuinely trying her best despite her anxiety.
Their training is led by a senior employee who's surprisingly clueless about how to train them. As the trio navigate the challenges of dealing with the public and a less-than-helpful supervisor, they struggle with the realities of their chosen profession.
Despite their initial disillusionment, they find ways to support each other and strive to make a positive impact, even in a seemingly mundane environment. The anime explores their daily lives, highlighting both the frustrations and the unexpected camaraderie they develop.