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Three different approaches to precisely describe the Adler function in the Euclidean regime at around 2 GeVs are available: dispersion relations based on the hadronic production data in e+e− annihilation, lattice simulations and perturbative QCD (pQCD). We make a comprehensive study of the perturbative approach, supplemented with the leading power corrections in the operator product expansion. All known contributions are included, with a careful assessment of uncertainties. The pQCD predictions are compared with the Adler functions extracted from Δ α QED had (Q2), using both the DHMZ compilation of e+e− data and published lattice results. Taking as input the FLAG value of αs, the pQCD Adler function turns out to be in good agreement with the lattice data, while the dispersive results lie systematically below them. Finally, we explore the sensitivity to αs of the direct comparison between the data-driven, lattice and QCD Euclidean Adler functions. The precision with which the renormalisation group equation can be tested is also evaluated. [ABSTRACT FROM AUTHOR]