| L(s) = 1 | + 2-s + 3·3-s + 4·4-s + 4·5-s + 3·6-s + 4·7-s + 7·8-s + 3·9-s + 4·10-s − 7·11-s + 12·12-s + 8·13-s + 4·14-s + 12·15-s + 14·16-s + 2·17-s + 3·18-s + 6·19-s + 16·20-s + 12·21-s − 7·22-s + 8·23-s + 21·24-s + 6·25-s + 8·26-s + 16·28-s − 29-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1.73·3-s + 2·4-s + 1.78·5-s + 1.22·6-s + 1.51·7-s + 2.47·8-s + 9-s + 1.26·10-s − 2.11·11-s + 3.46·12-s + 2.21·13-s + 1.06·14-s + 3.09·15-s + 7/2·16-s + 0.485·17-s + 0.707·18-s + 1.37·19-s + 3.57·20-s + 2.61·21-s − 1.49·22-s + 1.66·23-s + 4.28·24-s + 6/5·25-s + 1.56·26-s + 3.02·28-s − 0.185·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{16} \cdot 5^{8} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{16} \cdot 5^{8} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(37.92337952\) |
| \(L(\frac12)\) |
\(\approx\) |
\(37.92337952\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - p T + 2 p T^{2} - p^{2} T^{3} + p^{2} T^{4} - p^{3} T^{5} + 2 p^{3} T^{6} - p^{4} T^{7} + p^{4} T^{8} \) |
| 5 | \( ( 1 - T + T^{2} )^{4} \) |
| 7 | \( ( 1 - T + T^{2} )^{4} \) |
| good | 2 | \( ( 1 - 5 T + 13 T^{2} - 25 T^{3} + 39 T^{4} - 25 p T^{5} + 13 p^{2} T^{6} - 5 p^{3} T^{7} + p^{4} T^{8} )( 1 + p^{2} T + p^{2} T^{2} - 7 T^{3} - 21 T^{4} - 7 p T^{5} + p^{4} T^{6} + p^{5} T^{7} + p^{4} T^{8} ) \) |
| 11 | \( 1 + 7 T - 9 T^{2} - 72 T^{3} + 656 T^{4} + 2001 T^{5} - 7276 T^{6} + 844 T^{7} + 151647 T^{8} + 844 p T^{9} - 7276 p^{2} T^{10} + 2001 p^{3} T^{11} + 656 p^{4} T^{12} - 72 p^{5} T^{13} - 9 p^{6} T^{14} + 7 p^{7} T^{15} + p^{8} T^{16} \) |
| 13 | \( 1 - 8 T - 2 T^{2} + 50 T^{3} + 725 T^{4} - 1727 T^{5} - 11281 T^{6} + 7925 T^{7} + 170479 T^{8} + 7925 p T^{9} - 11281 p^{2} T^{10} - 1727 p^{3} T^{11} + 725 p^{4} T^{12} + 50 p^{5} T^{13} - 2 p^{6} T^{14} - 8 p^{7} T^{15} + p^{8} T^{16} \) |
| 17 | \( ( 1 - T + 64 T^{2} - 47 T^{3} + 1599 T^{4} - 47 p T^{5} + 64 p^{2} T^{6} - p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 19 | \( ( 1 - 3 T + 45 T^{2} - 168 T^{3} + 989 T^{4} - 168 p T^{5} + 45 p^{2} T^{6} - 3 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 23 | \( 1 - 8 T + 18 T^{2} - 240 T^{3} + 1445 T^{4} - 1872 T^{5} + 33674 T^{6} - 166760 T^{7} + 89244 T^{8} - 166760 p T^{9} + 33674 p^{2} T^{10} - 1872 p^{3} T^{11} + 1445 p^{4} T^{12} - 240 p^{5} T^{13} + 18 p^{6} T^{14} - 8 p^{7} T^{15} + p^{8} T^{16} \) |
| 29 | \( 1 + T - 61 T^{2} + 156 T^{3} + 1796 T^{4} - 8467 T^{5} - 22034 T^{6} + 147078 T^{7} + 433657 T^{8} + 147078 p T^{9} - 22034 p^{2} T^{10} - 8467 p^{3} T^{11} + 1796 p^{4} T^{12} + 156 p^{5} T^{13} - 61 p^{6} T^{14} + p^{7} T^{15} + p^{8} T^{16} \) |
| 31 | \( 1 - 94 T^{2} - 90 T^{3} + 4885 T^{4} + 5625 T^{5} - 188701 T^{6} - 84465 T^{7} + 6132709 T^{8} - 84465 p T^{9} - 188701 p^{2} T^{10} + 5625 p^{3} T^{11} + 4885 p^{4} T^{12} - 90 p^{5} T^{13} - 94 p^{6} T^{14} + p^{8} T^{16} \) |
| 37 | \( ( 1 + 21 T + 289 T^{2} + 2592 T^{3} + 18369 T^{4} + 2592 p T^{5} + 289 p^{2} T^{6} + 21 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 41 | \( 1 + 20 T + 126 T^{2} + 390 T^{3} + 5225 T^{4} + 43635 T^{5} + 118739 T^{6} + 1639535 T^{7} + 19690659 T^{8} + 1639535 p T^{9} + 118739 p^{2} T^{10} + 43635 p^{3} T^{11} + 5225 p^{4} T^{12} + 390 p^{5} T^{13} + 126 p^{6} T^{14} + 20 p^{7} T^{15} + p^{8} T^{16} \) |
| 43 | \( 1 - 7 T - 77 T^{2} + 160 T^{3} + 5570 T^{4} + 5177 T^{5} - 258466 T^{6} - 11150 T^{7} + 7688479 T^{8} - 11150 p T^{9} - 258466 p^{2} T^{10} + 5177 p^{3} T^{11} + 5570 p^{4} T^{12} + 160 p^{5} T^{13} - 77 p^{6} T^{14} - 7 p^{7} T^{15} + p^{8} T^{16} \) |
| 47 | \( 1 - 2 T - 58 T^{2} - 714 T^{3} + 2873 T^{4} + 817 p T^{5} + 291889 T^{6} - 1740327 T^{7} - 14193947 T^{8} - 1740327 p T^{9} + 291889 p^{2} T^{10} + 817 p^{4} T^{11} + 2873 p^{4} T^{12} - 714 p^{5} T^{13} - 58 p^{6} T^{14} - 2 p^{7} T^{15} + p^{8} T^{16} \) |
| 53 | \( ( 1 + 8 T + 46 T^{2} - 401 T^{3} - 3531 T^{4} - 401 p T^{5} + 46 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 59 | \( 1 + 19 T + p T^{2} - 876 T^{3} - 244 T^{4} + 106457 T^{5} + 689896 T^{6} + 11148 p T^{7} - 131107 p T^{8} + 11148 p^{2} T^{9} + 689896 p^{2} T^{10} + 106457 p^{3} T^{11} - 244 p^{4} T^{12} - 876 p^{5} T^{13} + p^{7} T^{14} + 19 p^{7} T^{15} + p^{8} T^{16} \) |
| 61 | \( 1 - 12 T - 49 T^{2} + 852 T^{3} + 1876 T^{4} - 9576 T^{5} - 372571 T^{6} + 33096 T^{7} + 27681967 T^{8} + 33096 p T^{9} - 372571 p^{2} T^{10} - 9576 p^{3} T^{11} + 1876 p^{4} T^{12} + 852 p^{5} T^{13} - 49 p^{6} T^{14} - 12 p^{7} T^{15} + p^{8} T^{16} \) |
| 67 | \( ( 1 - 11 T - 42 T^{2} - 319 T^{3} + 13973 T^{4} - 319 p T^{5} - 42 p^{2} T^{6} - 11 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 71 | \( ( 1 - 13 T + 248 T^{2} - 1951 T^{3} + 23005 T^{4} - 1951 p T^{5} + 248 p^{2} T^{6} - 13 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 73 | \( ( 1 + 4 T + 78 T^{2} + 545 T^{3} + 761 T^{4} + 545 p T^{5} + 78 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 79 | \( 1 - 24 T + 279 T^{2} - 2304 T^{3} + 10516 T^{4} + 43728 T^{5} - 930339 T^{6} + 9097068 T^{7} - 88845273 T^{8} + 9097068 p T^{9} - 930339 p^{2} T^{10} + 43728 p^{3} T^{11} + 10516 p^{4} T^{12} - 2304 p^{5} T^{13} + 279 p^{6} T^{14} - 24 p^{7} T^{15} + p^{8} T^{16} \) |
| 83 | \( 1 - 19 T + 8 T^{2} + 1653 T^{3} - 472 T^{4} - 87362 T^{5} - 374159 T^{6} - 2991816 T^{7} + 141658213 T^{8} - 2991816 p T^{9} - 374159 p^{2} T^{10} - 87362 p^{3} T^{11} - 472 p^{4} T^{12} + 1653 p^{5} T^{13} + 8 p^{6} T^{14} - 19 p^{7} T^{15} + p^{8} T^{16} \) |
| 89 | \( ( 1 - 15 T + 241 T^{2} - 1380 T^{3} + 17331 T^{4} - 1380 p T^{5} + 241 p^{2} T^{6} - 15 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 97 | \( 1 - 12 T - 33 T^{2} + 1956 T^{3} - 21452 T^{4} + 120264 T^{5} + 108549 T^{6} - 18621312 T^{7} + 276817383 T^{8} - 18621312 p T^{9} + 108549 p^{2} T^{10} + 120264 p^{3} T^{11} - 21452 p^{4} T^{12} + 1956 p^{5} T^{13} - 33 p^{6} T^{14} - 12 p^{7} T^{15} + p^{8} T^{16} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{16} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−5.15393369089035688179796604720, −5.12873096011274237501636197109, −5.09410324838251972442125888592, −4.96664802270684126937228166038, −4.91128189222529415259664333361, −4.68344626265241798834235830247, −4.17821570213085476392047490517, −4.02743460985799548239031572868, −3.79307443130603021475709983768, −3.75970132575257263536267884388, −3.45881547737444071391489839112, −3.39531383060288808263305151044, −3.32590314902189807505755629525, −3.30963002242333327822755348794, −2.96694711042788266459235065970, −2.78352596609215494262580098601, −2.60971089004785598970714895295, −2.23232450793004479345777780143, −2.13467894961581313023926891479, −2.04890801554905337656108617121, −1.80148891083579582944791580404, −1.62258224741643344565558709093, −1.60599060142200559942578632967, −1.15405195148561930777836847944, −0.983294803129153797982880801605,
0.983294803129153797982880801605, 1.15405195148561930777836847944, 1.60599060142200559942578632967, 1.62258224741643344565558709093, 1.80148891083579582944791580404, 2.04890801554905337656108617121, 2.13467894961581313023926891479, 2.23232450793004479345777780143, 2.60971089004785598970714895295, 2.78352596609215494262580098601, 2.96694711042788266459235065970, 3.30963002242333327822755348794, 3.32590314902189807505755629525, 3.39531383060288808263305151044, 3.45881547737444071391489839112, 3.75970132575257263536267884388, 3.79307443130603021475709983768, 4.02743460985799548239031572868, 4.17821570213085476392047490517, 4.68344626265241798834235830247, 4.91128189222529415259664333361, 4.96664802270684126937228166038, 5.09410324838251972442125888592, 5.12873096011274237501636197109, 5.15393369089035688179796604720
Plot not available for L-functions of degree greater than 10.
