Source : Webster's Revised Unabridged Dictionary (1913)

Righteously \Right"eous*ly\, adv. [AS. rightw[=i]sl[=i]ce.]
   In a righteous manner; as, to judge righteously.

Source : WordNet®

righteously
     adv : in a righteous manner; "righteously indignant" [ant: {unrighteously}]