 If you liked
Ginga Eiyuu Densetsu
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 ...then you might like
Taiyou no Kiba Dougram
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While LOGH is in a league of it's own, Dagram has a mature story with deep intrigue which steps into a political realm with some intersting criticism and actual real feel in which few anime series dare to go.
If you're into a bit of a mature storyline and don't mind the physics innacuracies or older animation, do give the other show a try.
Both of these shows take a serious approach to politics in comparison with most sci-fi shows. Dougram is the earlier show, and it shows - it's less polished and not as epic as LoGH, but that aside it's a very solid series with some good mech action and a plot which doesn't shy away from exploring the depths humanity sinks to.
Dougram may not have the same technical quality as LOGH, but it's the only work that comes close to it with themes.
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Recommended by Pipebomb_Teller
Fang of the Sun Dougram is the mecha version of Legend of the Galactic Heroes, and also a precursor to it in some way. They are both heavy political hard sci-fi shows, with many episodes and a strong storyline. Dougram is not as much well written and non-Manichean than LoGH but very close to it. The political focus, the strategic battles, the historical/documentary aspect, the realism of the deep worldbuilding and the very humans characters of Dougram strongly remind LoGH. Just replace huge space fleets battles by grounded guerrilla fight with mecha.
If you're a fan of LoGH, Fang of the Sun Dougram is the next
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