To serve in an official capacity — specifically the cultic service of priests and Levites, and the royal service of courtiers. A higher, more dignified word than ordinary labor; it implies honor in serving.
Priests sharath God; angels sharath as divine messengers; ministers sharath kings. Service to God is the highest form of sharath — an honor, not a burden.
Hebrews 1:14 echoes sharath with leitourgika (liturgical/ministerial spirits). Angels are divine ministers; believers are priests (1 Pet 2:9) — the whole redeemed community becomes a worshipping, serving body.