Grate the fresh ginger and combine it with 1 cup of filtered water and your sweetener in a saucepan. Heat gently over medium heat, stirring until the sweetener dissolves, about 5 minutes. Remove from heat and let it cool before straining out the ginger.

Roll the lemons on a countertop, cut in half, and juice them. Strain the juice to eliminate excess pulp, aiming for about 1 cup of juice.

In a large pitcher, combine the cooled ginger syrup, lemon juice, and remaining 3 cups of filtered water. Muddle the fresh mint leaves into the mixture, stir until combined, and adjust sweetness as desired.

Cover the pitcher and refrigerate for at least 1 hour. Serve over ice, garnished with lemon slices and a sprig of mint.