Luke 3:21-4:1-13.
Jesus had become by this time, a mature man, well-prepared physically and spiritually. It is time for the beginning of His public ministry. First he goes to John to be baptized by him. The three other Gospels give more detail of this event. Luke only includes what God did and said after the baptism itself. He is concerned about what God revealed about Jesus at the beginning of his public appearance.

3:21. Those of the Jewish people who had turned to God are not numbered, but a great many had by this time come to John. It was then that Jesus went and was baptized by John. By His submitting to this rite, the Sinless One took the sin of mankind upon Himself. He offers Himself as the Substitute, to be identified with and represent the sinful race of man. This was the outward and public sign that He accepted His work of redemption which began here and which was not to be completed until His suffering and death.

Jesus was praying in communion with God, when the heaven opened - a brief window to the majesty and glory of the father and Jesus. Jesus and also John (read John 1:32) saw the Holy Spirit descending from heaven in the form of a dove - a symbol of purity, innocence, beauty, and peace. This means Jesus was henceforth completely equipped to fulfill the role of Messiah and Redeemer - openly shown in public as a further sign of His designation as the Christ of God. His eternal Son-ship is further confirmed by a voice from heaven saying, "Thou art my beloved Son, in whom I am well pleased."

Verse 22. "in Thee I am well pleased" - in reference to everything Jesus had done up to this time - all within God's will. Now He was especially in God's will by submitting to baptism - officially and publicly taking on the work of redemption, ultimately leading to the Cross. In Christ Jesus, heaven has been opened to us, and the way prepared. We have been redeemed and become God's children by that Redemption.

Verses 23-38 - The Genealogical Table of Jesus. Having shown the background of Jesus and beginning of His public ministry, Luke chose to place the genealogical table here.

Verse 23. He mentions the age of Jesus (the only one to do so) as about 30 years old. Genealogical tables had been compiled and updated for many centuries. Public registers were kept, and many families also did.

Luke obtained a copy of the genealogical table of Mary's father Heli. Since it was not customary to insert the name of a woman in such a list he added (as was supposed) the son of Joseph - in Luke 1 & 2 he has recorded that Jesus was solely the son of Mary. In Matthew we find the family tree of Joseph who was legally Jesus' father (though only stepfather, actually). This table is quite different from Luke's. Yet, it clearly establishes Davidic descent legally.

As Luke had been writing for Romans and Greeks, Jesus' Davidic descent was not essential. Luke wanted to show Jesus' actual human descent through David, Abraham, to Adam the son of God. That Mary was a descendant of David was never disputed during the early centuries.

Verses 24-38. By going all the way back to Adam, Luke shows Jesus' ancestry not only from the Jewish line to Abraham, but before the existence of that line. He shows the family tree of all mankind which began with Adam - the son of God in that he was created by God as a mature man. Jesus becomes the second Adam - born of woman but conceived by the Holy Spirit. Many books have been written that go into the details of these lists identifying as many of the unfamiliar names and their places on the lists.