Paul use both terms and there aren't any crossnotes.
Romans 8:1 - 9:33
ESV - 1 There is therefore now no condemnation for those who are in Christ Jesus. 2 For the law of the Spirit of life has set you free in Christ Jesus from the law of sin and death.
For follow-up discussion and general commentary on the topic. Comments are sorted chronologically.
It is my opinion that the Spirit of God, sometimes rendered as spirit of God, lower case "s" and Holy Spirit, in most scripture verses, have a different meaning; God is one, yet three persons (Trinitarian doctrine). But in Mt. 3:16, Lk. 3:22, and Jn. 1:33, we seem to have an exception to this rule. Matthew uses "Spirit of God," while Luke uses "Holy Spirit," and John, retelling the events of the baptism of Jesus, uses "Spirit of God." Matthew and John both use "Spirit" (without including Holy) with an upper case "S" which does not always mean or imply Holy Spirit in other verses of the N.T. This can be at times somewhat confusing to the average reader, especially when reading the O.T., where Holy Spirit, Heb: "Ruach HaKodesh," is found only 3 times (Ps. 1:11; Is. 63:10,11). It is essential to always look at the original language and see what was written before translators produced their works of scripture.