What does the Gospel of Luke present Jesus as?

The special emphasis of Luke is the humanity of Jesus. While each of the Gospels were intended for all mankind, Matthew seems to have had in immediate view the Jews; Mark, the Romans; and Luke, the Greeks. Greek civilization represented culture, philosophy, wisdom, reason, beauty, and education. Therefore, to appeal to the thoughtful, cultured, philosophic Greek mind, Luke, in a complete, orderly, and classical story, depicts the beauty and perfection of Jesus, the ideal, universal man.

Another Answer:

Luke is the only Gospel writer who did not know Jesus like the other 3 disciples. He writes the only Gospel with a sequel - Book of Acts. He was an educated man - a physician (Colossians 4:14).

For Luke, Jesus is the promised Messiah (1:31-35), the Son of God (9:35), the Servant through whom God works (4:16-18), and the Lord who sits at God's right hand exerting His authority and giving His Spirit to anyone who repents and believes (compare 22:69 with Acts 2:30-36).

Luke emphasizes the Old Testament and its promises of a Messiah in the Book's first two chapters while the third chapter demonstrates that Jesus is the Messiah who can resist the Evil One. Luke goes on introducing Jesus' power and teachings. By the time we come to chapter 24, His disciples should realize Jesus is the fulfillment of the Law, the Prophets, and the Psalms. (v 43-49).

It is clear Luke's Gospel centers on God's plan to provide salvation to the world. It closes with the anticipation of the spead of the gospel which is recorded in Acts.