| L(s) = 1 | − 2·2-s + 5·4-s − 2·8-s − 6·9-s + 16·11-s + 2·16-s + 12·18-s − 32·22-s + 8·23-s − 18·25-s − 22·29-s + 12·32-s − 30·36-s + 6·37-s − 6·43-s + 80·44-s − 16·46-s + 36·50-s − 28·53-s + 44·58-s − 17·64-s + 76·71-s + 12·72-s − 12·74-s + 6·79-s + 24·81-s + 12·86-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 5/2·4-s − 0.707·8-s − 2·9-s + 4.82·11-s + 1/2·16-s + 2.82·18-s − 6.82·22-s + 1.66·23-s − 3.59·25-s − 4.08·29-s + 2.12·32-s − 5·36-s + 0.986·37-s − 0.914·43-s + 12.0·44-s − 2.35·46-s + 5.09·50-s − 3.84·53-s + 5.77·58-s − 2.12·64-s + 9.01·71-s + 1.41·72-s − 1.39·74-s + 0.675·79-s + 8/3·81-s + 1.29·86-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{24} \cdot 7^{24}\right)^{s/2} \, \Gamma_{\C}(s)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{24} \cdot 7^{24}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.146320587\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.146320587\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + 2 p T^{2} + 4 p T^{4} + p^{2} T^{6} + 4 p^{3} T^{8} + 2 p^{5} T^{10} + p^{6} T^{12} \) |
| 7 | \( 1 \) |
| good | 2 | \( ( 1 + T - T^{2} - p^{2} T^{3} - 3 T^{4} + p T^{5} + 13 T^{6} + p^{2} T^{7} - 3 p^{2} T^{8} - p^{5} T^{9} - p^{4} T^{10} + p^{5} T^{11} + p^{6} T^{12} )^{2} \) |
| 5 | \( ( 1 + 9 T^{2} + 63 T^{4} + 349 T^{6} + 63 p^{2} T^{8} + 9 p^{4} T^{10} + p^{6} T^{12} )^{2} \) |
| 11 | \( ( 1 - 4 T + 32 T^{2} - 87 T^{3} + 32 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} )^{4} \) |
| 13 | \( 1 - 3 p T^{2} + 51 p T^{4} - 6584 T^{6} + 5157 p T^{8} - 102645 p T^{10} + 22407342 T^{12} - 102645 p^{3} T^{14} + 5157 p^{5} T^{16} - 6584 p^{6} T^{18} + 51 p^{9} T^{20} - 3 p^{11} T^{22} + p^{12} T^{24} \) |
| 17 | \( 1 - 18 T^{2} + 108 T^{4} + 1706 T^{6} - 66114 T^{8} + 139734 T^{10} + 15553959 T^{12} + 139734 p^{2} T^{14} - 66114 p^{4} T^{16} + 1706 p^{6} T^{18} + 108 p^{8} T^{20} - 18 p^{10} T^{22} + p^{12} T^{24} \) |
| 19 | \( 1 - 39 T^{2} + 510 T^{4} + 31 p T^{6} - 123201 T^{8} + 3156120 T^{10} - 72752079 T^{12} + 3156120 p^{2} T^{14} - 123201 p^{4} T^{16} + 31 p^{7} T^{18} + 510 p^{8} T^{20} - 39 p^{10} T^{22} + p^{12} T^{24} \) |
| 23 | \( ( 1 - 2 T + 44 T^{2} - 33 T^{3} + 44 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} )^{4} \) |
| 29 | \( ( 1 + 11 T + 20 T^{2} + 13 T^{3} + 1233 T^{4} + 262 T^{5} - 47411 T^{6} + 262 p T^{7} + 1233 p^{2} T^{8} + 13 p^{3} T^{9} + 20 p^{4} T^{10} + 11 p^{5} T^{11} + p^{6} T^{12} )^{2} \) |
| 31 | \( 1 - 57 T^{2} - 678 T^{4} + 7483 T^{6} + 5054805 T^{8} - 2471706 p T^{10} - 2776215 p^{2} T^{12} - 2471706 p^{3} T^{14} + 5054805 p^{4} T^{16} + 7483 p^{6} T^{18} - 678 p^{8} T^{20} - 57 p^{10} T^{22} + p^{12} T^{24} \) |
| 37 | \( ( 1 - 3 T - 78 T^{2} + 237 T^{3} + 3603 T^{4} - 6456 T^{5} - 129067 T^{6} - 6456 p T^{7} + 3603 p^{2} T^{8} + 237 p^{3} T^{9} - 78 p^{4} T^{10} - 3 p^{5} T^{11} + p^{6} T^{12} )^{2} \) |
| 41 | \( 1 - 84 T^{2} + 2334 T^{4} + 75506 T^{6} - 5470866 T^{8} - 18588942 T^{10} + 7812254391 T^{12} - 18588942 p^{2} T^{14} - 5470866 p^{4} T^{16} + 75506 p^{6} T^{18} + 2334 p^{8} T^{20} - 84 p^{10} T^{22} + p^{12} T^{24} \) |
| 43 | \( ( 1 + 3 T - 96 T^{2} - 255 T^{3} + 5655 T^{4} + 8382 T^{5} - 250477 T^{6} + 8382 p T^{7} + 5655 p^{2} T^{8} - 255 p^{3} T^{9} - 96 p^{4} T^{10} + 3 p^{5} T^{11} + p^{6} T^{12} )^{2} \) |
| 47 | \( 1 - 99 T^{2} + 486 T^{4} - 19399 T^{6} + 21504483 T^{8} - 613600884 T^{10} - 14184900399 T^{12} - 613600884 p^{2} T^{14} + 21504483 p^{4} T^{16} - 19399 p^{6} T^{18} + 486 p^{8} T^{20} - 99 p^{10} T^{22} + p^{12} T^{24} \) |
| 53 | \( ( 1 + 14 T + 26 T^{2} - 62 T^{3} + 2796 T^{4} - 5384 T^{5} - 293669 T^{6} - 5384 p T^{7} + 2796 p^{2} T^{8} - 62 p^{3} T^{9} + 26 p^{4} T^{10} + 14 p^{5} T^{11} + p^{6} T^{12} )^{2} \) |
| 59 | \( 1 - 171 T^{2} + 10764 T^{4} - 658021 T^{6} + 63198909 T^{8} - 3185695998 T^{10} + 106058651361 T^{12} - 3185695998 p^{2} T^{14} + 63198909 p^{4} T^{16} - 658021 p^{6} T^{18} + 10764 p^{8} T^{20} - 171 p^{10} T^{22} + p^{12} T^{24} \) |
| 61 | \( 1 - 102 T^{2} - 2514 T^{4} + 108094 T^{6} + 46094616 T^{8} - 815218740 T^{10} - 153775720821 T^{12} - 815218740 p^{2} T^{14} + 46094616 p^{4} T^{16} + 108094 p^{6} T^{18} - 2514 p^{8} T^{20} - 102 p^{10} T^{22} + p^{12} T^{24} \) |
| 67 | \( ( 1 - 90 T^{2} + 706 T^{3} + 2070 T^{4} - 31770 T^{5} + 183435 T^{6} - 31770 p T^{7} + 2070 p^{2} T^{8} + 706 p^{3} T^{9} - 90 p^{4} T^{10} + p^{6} T^{12} )^{2} \) |
| 71 | \( ( 1 - 19 T + 329 T^{2} - 2925 T^{3} + 329 p T^{4} - 19 p^{2} T^{5} + p^{3} T^{6} )^{4} \) |
| 73 | \( 1 - 363 T^{2} + 72438 T^{4} - 10402079 T^{6} + 1182976731 T^{8} - 111093122556 T^{10} + 8799155948049 T^{12} - 111093122556 p^{2} T^{14} + 1182976731 p^{4} T^{16} - 10402079 p^{6} T^{18} + 72438 p^{8} T^{20} - 363 p^{10} T^{22} + p^{12} T^{24} \) |
| 79 | \( ( 1 - 3 T - 150 T^{2} + 257 T^{3} + 11619 T^{4} - 1710 T^{5} - 932601 T^{6} - 1710 p T^{7} + 11619 p^{2} T^{8} + 257 p^{3} T^{9} - 150 p^{4} T^{10} - 3 p^{5} T^{11} + p^{6} T^{12} )^{2} \) |
| 83 | \( 1 - 270 T^{2} + 31896 T^{4} - 3065782 T^{6} + 318136230 T^{8} - 24401026026 T^{10} + 1632699460815 T^{12} - 24401026026 p^{2} T^{14} + 318136230 p^{4} T^{16} - 3065782 p^{6} T^{18} + 31896 p^{8} T^{20} - 270 p^{10} T^{22} + p^{12} T^{24} \) |
| 89 | \( 1 - 288 T^{2} + 45954 T^{4} - 3797734 T^{6} + 88086726 T^{8} + 23906597562 T^{10} - 3299038805433 T^{12} + 23906597562 p^{2} T^{14} + 88086726 p^{4} T^{16} - 3797734 p^{6} T^{18} + 45954 p^{8} T^{20} - 288 p^{10} T^{22} + p^{12} T^{24} \) |
| 97 | \( 1 - 471 T^{2} + 121776 T^{4} - 22400861 T^{6} + 3247635573 T^{8} - 392084795286 T^{10} + 40704346255641 T^{12} - 392084795286 p^{2} T^{14} + 3247635573 p^{4} T^{16} - 22400861 p^{6} T^{18} + 121776 p^{8} T^{20} - 471 p^{10} T^{22} + p^{12} T^{24} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{24} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−3.70877599769213204040028168154, −3.65319047261123497475738269340, −3.63558629890152303414607896784, −3.52499583555760908786661765279, −3.37275197335692125292191043693, −3.30539016556965815787328486536, −3.02103317421056553031080166242, −2.85477615780255834684628357904, −2.70233070773490066070666169795, −2.67652209106544895360883273411, −2.57018625239071644624316434413, −2.44397601713889587336034743851, −2.19004695094396587508902487934, −2.16646548128725599645950592915, −1.99049686683857454453441801538, −1.84477717159881129172498998362, −1.74566544168349118372809125927, −1.58851712260859333159429928568, −1.55742785345645314644683549015, −1.54756287828522052360527471405, −1.10231841662325580025395473489, −1.09089056097288579475028094073, −1.01042773970483991256133649831, −0.56352934482931732821145294547, −0.20419701791427244713212913465,
0.20419701791427244713212913465, 0.56352934482931732821145294547, 1.01042773970483991256133649831, 1.09089056097288579475028094073, 1.10231841662325580025395473489, 1.54756287828522052360527471405, 1.55742785345645314644683549015, 1.58851712260859333159429928568, 1.74566544168349118372809125927, 1.84477717159881129172498998362, 1.99049686683857454453441801538, 2.16646548128725599645950592915, 2.19004695094396587508902487934, 2.44397601713889587336034743851, 2.57018625239071644624316434413, 2.67652209106544895360883273411, 2.70233070773490066070666169795, 2.85477615780255834684628357904, 3.02103317421056553031080166242, 3.30539016556965815787328486536, 3.37275197335692125292191043693, 3.52499583555760908786661765279, 3.63558629890152303414607896784, 3.65319047261123497475738269340, 3.70877599769213204040028168154
Plot not available for L-functions of degree greater than 10.
