Our aim was to determine the initial Li content of two clusters of similar metallicity but very different ages, the old open cluster NGC 2243 and the metal-rich globular cluster NGC 104. We compared the lithium abundances derived for a large sample of stars (from the turn-off to the red giant branch) in each cluster. For NGC 2243 the Li abundances are from the catalogues released by the Gaia-ESO Public Spectroscopic Survey, while for NGC 104 we measured the Li abundance using FLAMES/GIRAFFE spectra, which include archival data and new observations. We took the initial Li of NGC 2243 to be the lithium measured in stars on the hot side of the Li dip. We used the difference between the initial abundances and the post first dredge-up Li values of NGC 2243, and by adding this amount to the post first dredge-up stars of NGC~104 we were able to infer the initial Li of this cluster. Moreover, we compared our observational results to the predictions of theoretical stellar models for the difference between the initial Li abundance and that after the first dredge-up. The initial lithium content of NGC 2243 was found to be A(Li)_i = 2.85dex by taking the average Li abundance measured from the five hottest stars with the highest lithium abundance. This value is 1.69 dex higher than the lithium abundance derived in post first dredge-up stars. By adding this number to the lithium abundance derived in the post first dredge-up stars in NGC~104, we infer a lower limit of its initial lithium content of A(Li)_i= 2.30dex. Stellar models predict similar values. Therefore, our result offers important insights for further theoretical developments.
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