The third pattern of the C Major scale uses the G shape open chord, When looking at the open G chord the root notes are on the open G string and the 3rd fret top and bottom E strings.
G
Reminder: The notes used in the C Major scale are C D E F G A B, there are no sharps or flats in the C Major scale.
So if we take a look at the scale on the fretboard, you will see something like the following pattern… the root notes are marked with the “X“…
| | $3.4 | $1.5.$2.5.$3.X.$4.5.$5.5.$6.5 | $2.6 | $1.7.$3.7.$4.7.$5.7.$6.7 | $1.X.$2.8.$5.8.$6.X |
Note: In the diagram above it’s the fret numbers that are indicated not the scale degrees, and it isn’t the way you would normally write out a scale using TAB, it’s just to illustrate the fingering pattern of the scale. 
When playing this scale on the guitar, I would normally start on the lowest pitch root note, 6th string 8th fret, and play up to the highest pitch note of the scale, 1st string 8th fret, which is also one of the root notes within the scale.
| $6.8 $5.5 7 8 $4.5 7 $3.4 5 7 $2.5 6 8 $1.5 7 8 |
Before descending back to the lowest pitch note of the scale, the A on the 5th fret 6th string, then back up in pitch to the C root note on the 8th fret 6th string.
| $1.8 7 5 $2.8 6 5 $3.7 5 4 $4.7 5 $5.8 7 5 $6.8 7 5 | $6.5 7 8 |