Legendre Polynomial # Zero # Value of Zero 0 0 -1.000000000000000 0 1 1.000000000000000 1 0 -1.000000000000000 1 1 0.000000000000000 1 2 1.000000000000000 2 0 -1.000000000000000 2 1 -0.5773502691898591 2 2 0.5773502691898591 2 3 1.000000000000000 3 0 -1.000000000000000 3 1 -0.7745966692414834 3 2 0.000000000000000 3 3 0.7745966692414834 3 4 1.000000000000000 4 0 -1.000000000000000 4 1 -0.8611363115940526 4 2 -0.3399810435849411 4 3 0.3399810435849411 4 4 0.8611363115940526 4 5 1.000000000000000 5 0 -1.000000000000000 5 1 -0.9061798459386640 5 2 -0.5384693101056831 5 3 0.000000000000000 5 4 0.5384693101056831 5 5 0.9061798459386640 5 6 1.000000000000000 6 0 -1.000000000000000 6 1 -0.9324695142031846 6 2 -0.6612093864662645 6 3 -0.2386191860831969 6 4 0.2386191860831969 6 5 0.6612093864662645 6 6 0.9324695142031846 6 7 1.000000000000000 7 0 -1.000000000000000 7 1 -0.9491079123427585 7 2 -0.7415311855994028 7 3 -0.4058451513773972 7 4 0.000000000000000 7 5 0.4058451513773972 7 6 0.7415311855994028 7 7 0.9491079123427585 7 8 1.000000000000000 8 0 -1.000000000000000 8 1 -0.9602898564975474 8 2 -0.7966664774136267 8 3 -0.5255324099166152 8 4 -0.1834346424956501 8 5 0.1834346424956501 8 6 0.5255324099166152 8 7 0.7966664774136267 8 8 0.9602898564975474 8 9 1.000000000000000 9 0 -1.000000000000000 9 1 -0.9681602395076261 9 2 -0.8360311073266358 9 3 -0.6133714327005904 9 4 -0.3242534234038089 9 5 0.000000000000000 9 6 0.3242534234038089 9 7 0.6133714327005904 9 8 0.8360311073266358 9 9 0.9681602395076261 9 10 1.000000000000000 10 0 -1.000000000000000 10 1 -0.9739065285171717 10 2 -0.8650633666889959 10 3 -0.6794095682990244 10 4 -0.4333953941292938 10 5 -0.1488743389816514 10 6 0.1488743389816514 10 7 0.4333953941292938 10 8 0.6794095682990244 10 9 0.8650633666889959 10 10 0.9739065285171717 10 11 1.000000000000000 11 0 -1.000000000000000 11 1 -0.9782286581460570 11 2 -0.8870625997681056 11 3 -0.7301520055740494 11 4 -0.5190961292068118 11 5 -0.2695431559523498 11 6 0.000000000000000 11 7 0.2695431559523498 11 8 0.5190961292068118 11 9 0.7301520055740494 11 10 0.8870625997681056 11 11 0.9782286581460570 11 12 1.000000000000000 12 0 -1.000000000000000 12 1 -0.9815606342467192 12 2 -0.9041172563704751 12 3 -0.7699026741943047 12 4 -0.5873179542866175 12 5 -0.3678314989981802 12 6 -0.1252334085114699 12 7 0.1252334085114699 12 8 0.3678314989981802 12 9 0.5873179542866175 12 10 0.7699026741943047 12 11 0.9041172563704751 12 12 0.9815606342467192 12 13 1.000000000000000 13 0 -1.000000000000000 13 1 -0.9841830547186112 13 2 -0.9175983992229779 13 3 -0.8015780907333366 13 4 -0.6423493394403402 13 5 -0.4484927510364469 13 6 -0.2304583159551406 13 7 0.000000000000000 13 8 0.2304583159551406 13 9 0.4484927510364469 13 10 0.6423493394403402 13 11 0.8015780907333366 13 12 0.9175983992229779 13 13 0.9841830547186112 13 14 1.000000000000000 14 0 -1.000000000000000 14 1 -0.9862838086968123 14 2 -0.9284348836635735 14 3 -0.8272013150698311 14 4 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0.4580167776572274 16 12 0.6178762444026438 16 13 0.7554044083550030 16 14 0.8656312023878318 16 15 0.9445750230732326 16 16 0.9894009349916499 16 17 1.000000000000000 17 0 -1.000000000000000 17 1 -0.9905754753144254 17 2 -0.9506755217687678 17 3 -0.8802391537270668 17 4 -0.7815140038968014 17 5 -0.6576711592166907 17 6 -0.5126905370864787 17 7 -0.3512317634540652 17 8 -0.1784841814958479 17 9 0.000000000000000 17 10 0.1784841814958479 17 11 0.3512317634540652 17 12 0.5126905370864787 17 13 0.6576711592166907 17 14 0.7815140038968014 17 15 0.8802391537270668 17 16 0.9506755217687678 17 17 0.9905754753144254 17 18 1.000000000000000 18 0 -1.000000000000000 18 1 -0.9915651684209309 18 2 -0.9558239495713977 18 3 -0.8926024664975557 18 4 -0.8037049589725231 18 5 -0.6916870430603532 18 6 -0.5597708310739475 18 7 -0.4117511614628426 18 8 -0.2518862256915057 18 9 -0.08477501304174115 18 10 0.08477501304174115 18 11 0.2518862256915057 18 12 0.4117511614628426 18 13 0.5597708310739475 18 14 0.6916870430603532 18 15 0.8037049589725231 18 16 0.8926024664975557 18 17 0.9558239495713977 18 18 0.9915651684209309 18 19 1.000000000000000 19 0 -1.000000000000000 19 1 -0.9924068438435844 19 2 -0.9602081521348300 19 3 -0.9031559036148179 19 4 -0.8227146565371428 19 5 -0.7209661773352294 19 6 -0.6005453046617087 19 7 -0.4645707413759609 19 8 -0.3165640999636300 19 9 -0.1603586456402254 19 10 0.000000000000000 19 11 0.1603586456402254 19 12 0.3165640999636300 19 13 0.4645707413759609 19 14 0.6005453046617087 19 15 0.7209661773352294 19 16 0.8227146565371428 19 17 0.9031559036148179 19 18 0.9602081521348300 19 19 0.9924068438435844 19 20 1.000000000000000 20 0 -1.000000000000000 20 1 -0.9931285991850985 20 2 -0.9639719272779138 20 3 -0.9122344282513259 20 4 -0.8391169718222192 20 5 -0.7463319064601509 20 6 -0.6360536807265150 20 7 -0.5108670019508371 20 8 -0.3737060887154195 20 9 -0.2277858511416470 20 10 -0.07652652113351606 20 11 0.07652652113351606 20 12 0.2277858511416470 20 13 0.3737060887154195 20 14 0.5108670019508371 20 15 0.6360536807265150 20 16 0.7463319064601509 20 17 0.8391169718222192 20 18 0.9122344282513259 20 19 0.9639719272779138 20 20 0.9931285991850985 20 21 1.000000000000000