It is worth noting that Paul’s view of prayer was Trinitarian. He refers to the lordship of Jesus, to the love of the Spirit, and prayers to the Father. This is a reminder of how Christians are to think of God.
With regard to the lordship of Jesus, we can think about two features connected to the position that Jesus occupies as sovereign Lord. First, as Lord, he commanded his people to pray. This is so obvious that we are liable to forget that prayer is what he expects from his people. Second, as Lord he has been exalted to answer the prayers of his people. He said so in the upper room when he encouraged his disciples to pray to him after he had left them and returned to heaven. So if he has commanded us to pray and has encouraged us to pray, we have reasons to engage in prayer.
Paul also mentions the love of the Spirit here. Prayer is one way in which the Spirit shows his love as he enables us to pray. It may be the case that Paul here is referring to the double intercession that he mentions in Romans 8 when he wrote that the Jesus intercedes for us in heaven and the Spirit intercedes for us in our hearts with groans too deep for words.
And we can see that Paul expected the Christians in Rome to address their prayers for him to God the Father. Inevitably by doing so, they would recognise that they and the apostle belonged to the same family and that mutual prayer is an expression of brotherly love. Paul is specific about what he wants them to pray for, an example of asking others to pray about particular areas in our lives.