A group by subquery to get the max time_stamp for all groups by serial_number joined with the original table.
You can write it this way, too, without WITH clause:
SELECT t.*
FROM ( SELECT serial_number, MAX (time_stamp) AS GroupMaxTimeStamp
FROM tableName
WHERE ERROR_CODE = 3
GROUP BY serial_number) g,
tableName t
WHERE t.serial_number = g.serial_number AND ERROR_CODE = 3
ORDER BY GroupMaxTimeStamp DESC, TIME_STAMP DESC;
You selecting the table twice, once for grouping and one to join with the grouping.
If its a very large table and not so many rows with error_code=3 then perhaps it is more efficient to filter the error_code first and then do the rest:
WITH tableWithErrorCode3
AS ( SELECT serial_number, time_stamp
FROM tableName
WHERE ERROR_CODE = 3)
SELECT t.*
FROM ( SELECT serial_number, MAX (time_stamp) AS GroupMaxTimeStamp