  We study the three-loop Euler-Heisenberg Lagrangian in spinor quantum
electrodynamics in 1+1 dimensions. In this first part we calculate the
one-fermion-loop contribution, applying both standard Feynman diagrams and the
worldline formalism which leads to two different representations in terms of
fourfold Schwinger-parameter integrals. Unlike the diagram calculation, the
worldline approach allows one to combine the planar and the non-planar
contributions to the Lagrangian. Our main interest is in the asymptotic
behaviour of the weak-field expansion coefficients of this Lagrangian, for
which a non-perturbative prediction has been obtained in previous work using
worldline instantons and Borel analysis. We develop algorithms for the
calculation of the weak-field expansions coefficients that, in principle, allow
their calculation to arbitrary order. Here for the non-planar contribution we
make essential use of the polynomial invariants of the dihedral group D4 in
Schwinger parameter space to keep the expressions manageable. As expected on
general grounds, the coefficients are of the form r1+r2*zeta(3) with rational
numbers r1, r2. We compute the first two coefficients analytically, and four
more by numerical integration.
