| L(s) = 1 | + 2-s − 4·3-s + 2·4-s − 4·5-s − 4·6-s + 7·7-s + 5·8-s + 6·9-s − 4·10-s − 11-s − 8·12-s + 20·13-s + 7·14-s + 16·15-s + 8·16-s + 4·17-s + 6·18-s − 13·19-s − 8·20-s − 28·21-s − 22-s − 3·23-s − 20·24-s + 17·25-s + 20·26-s + 14·28-s + 14·29-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 2.30·3-s + 4-s − 1.78·5-s − 1.63·6-s + 2.64·7-s + 1.76·8-s + 2·9-s − 1.26·10-s − 0.301·11-s − 2.30·12-s + 5.54·13-s + 1.87·14-s + 4.13·15-s + 2·16-s + 0.970·17-s + 1.41·18-s − 2.98·19-s − 1.78·20-s − 6.11·21-s − 0.213·22-s − 0.625·23-s − 4.08·24-s + 17/5·25-s + 3.92·26-s + 2.64·28-s + 2.59·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 7^{8} \cdot 17^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 7^{8} \cdot 17^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(6.477299907\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.477299907\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( ( 1 + T + T^{2} )^{4} \) |
| 7 | \( 1 - p T + 40 T^{2} - 149 T^{3} + 461 T^{4} - 149 p T^{5} + 40 p^{2} T^{6} - p^{4} T^{7} + p^{4} T^{8} \) |
| 17 | \( ( 1 - T + T^{2} )^{4} \) |
| good | 2 | \( 1 - T - T^{2} - p T^{3} + T^{4} + 3 p T^{5} + 5 p T^{6} - p^{4} T^{7} - 7 T^{8} - p^{5} T^{9} + 5 p^{3} T^{10} + 3 p^{4} T^{11} + p^{4} T^{12} - p^{6} T^{13} - p^{6} T^{14} - p^{7} T^{15} + p^{8} T^{16} \) |
| 5 | \( 1 + 4 T - T^{2} - 28 T^{3} - 4 p T^{4} + 24 p T^{5} + 41 p T^{6} - 344 T^{7} - 1681 T^{8} - 344 p T^{9} + 41 p^{3} T^{10} + 24 p^{4} T^{11} - 4 p^{5} T^{12} - 28 p^{5} T^{13} - p^{6} T^{14} + 4 p^{7} T^{15} + p^{8} T^{16} \) |
| 11 | \( 1 + T - 37 T^{2} - 16 T^{3} + 820 T^{4} + 129 T^{5} - 13058 T^{6} - 524 T^{7} + 161687 T^{8} - 524 p T^{9} - 13058 p^{2} T^{10} + 129 p^{3} T^{11} + 820 p^{4} T^{12} - 16 p^{5} T^{13} - 37 p^{6} T^{14} + p^{7} T^{15} + p^{8} T^{16} \) |
| 13 | \( ( 1 - 10 T + 76 T^{2} - 365 T^{3} + 1535 T^{4} - 365 p T^{5} + 76 p^{2} T^{6} - 10 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 19 | \( 1 + 13 T + 54 T^{2} + 9 p T^{3} + 1338 T^{4} + 3402 T^{5} - 19555 T^{6} - 92056 T^{7} - 147879 T^{8} - 92056 p T^{9} - 19555 p^{2} T^{10} + 3402 p^{3} T^{11} + 1338 p^{4} T^{12} + 9 p^{6} T^{13} + 54 p^{6} T^{14} + 13 p^{7} T^{15} + p^{8} T^{16} \) |
| 23 | \( 1 + 3 T - 74 T^{2} - 135 T^{3} + 3526 T^{4} + 3630 T^{5} - 119891 T^{6} - 29730 T^{7} + 3214381 T^{8} - 29730 p T^{9} - 119891 p^{2} T^{10} + 3630 p^{3} T^{11} + 3526 p^{4} T^{12} - 135 p^{5} T^{13} - 74 p^{6} T^{14} + 3 p^{7} T^{15} + p^{8} T^{16} \) |
| 29 | \( ( 1 - 7 T + 92 T^{2} - 549 T^{3} + 3801 T^{4} - 549 p T^{5} + 92 p^{2} T^{6} - 7 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 31 | \( 1 - 6 T - 88 T^{2} + 278 T^{3} + 6607 T^{4} - 10769 T^{5} - 302769 T^{6} + 120605 T^{7} + 10874553 T^{8} + 120605 p T^{9} - 302769 p^{2} T^{10} - 10769 p^{3} T^{11} + 6607 p^{4} T^{12} + 278 p^{5} T^{13} - 88 p^{6} T^{14} - 6 p^{7} T^{15} + p^{8} T^{16} \) |
| 37 | \( 1 - 6 T - 100 T^{2} + 344 T^{3} + 7711 T^{4} - 12680 T^{5} - 413472 T^{6} + 179198 T^{7} + 17368056 T^{8} + 179198 p T^{9} - 413472 p^{2} T^{10} - 12680 p^{3} T^{11} + 7711 p^{4} T^{12} + 344 p^{5} T^{13} - 100 p^{6} T^{14} - 6 p^{7} T^{15} + p^{8} T^{16} \) |
| 41 | \( ( 1 - 26 T + 335 T^{2} - 2856 T^{3} + 19581 T^{4} - 2856 p T^{5} + 335 p^{2} T^{6} - 26 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 43 | \( ( 1 + 4 T + 88 T^{2} - 59 T^{3} + 2941 T^{4} - 59 p T^{5} + 88 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 47 | \( 1 - 6 T - 134 T^{2} + 366 T^{3} + 13369 T^{4} - 13905 T^{5} - 897527 T^{6} + 316533 T^{7} + 45528901 T^{8} + 316533 p T^{9} - 897527 p^{2} T^{10} - 13905 p^{3} T^{11} + 13369 p^{4} T^{12} + 366 p^{5} T^{13} - 134 p^{6} T^{14} - 6 p^{7} T^{15} + p^{8} T^{16} \) |
| 53 | \( 1 + 15 T + 49 T^{2} - 102 T^{3} - 800 T^{4} - 22371 T^{5} - 194672 T^{6} - 558180 T^{7} - 1841867 T^{8} - 558180 p T^{9} - 194672 p^{2} T^{10} - 22371 p^{3} T^{11} - 800 p^{4} T^{12} - 102 p^{5} T^{13} + 49 p^{6} T^{14} + 15 p^{7} T^{15} + p^{8} T^{16} \) |
| 59 | \( 1 + 15 T + 34 T^{2} - 915 T^{3} - 9902 T^{4} - 68490 T^{5} - 215111 T^{6} + 4539300 T^{7} + 67748017 T^{8} + 4539300 p T^{9} - 215111 p^{2} T^{10} - 68490 p^{3} T^{11} - 9902 p^{4} T^{12} - 915 p^{5} T^{13} + 34 p^{6} T^{14} + 15 p^{7} T^{15} + p^{8} T^{16} \) |
| 61 | \( 1 - 11 T - 69 T^{2} + 348 T^{3} + 8430 T^{4} + 7773 T^{5} - 627598 T^{6} + 598580 T^{7} + 19487961 T^{8} + 598580 p T^{9} - 627598 p^{2} T^{10} + 7773 p^{3} T^{11} + 8430 p^{4} T^{12} + 348 p^{5} T^{13} - 69 p^{6} T^{14} - 11 p^{7} T^{15} + p^{8} T^{16} \) |
| 67 | \( 1 - 8 T + 39 T^{2} + 1032 T^{3} - 15024 T^{4} + 114864 T^{5} + 5705 T^{6} - 7844440 T^{7} + 105057207 T^{8} - 7844440 p T^{9} + 5705 p^{2} T^{10} + 114864 p^{3} T^{11} - 15024 p^{4} T^{12} + 1032 p^{5} T^{13} + 39 p^{6} T^{14} - 8 p^{7} T^{15} + p^{8} T^{16} \) |
| 71 | \( ( 1 - 15 T + 329 T^{2} - 3054 T^{3} + 36105 T^{4} - 3054 p T^{5} + 329 p^{2} T^{6} - 15 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 73 | \( 1 + 3 T - 52 T^{2} - 1525 T^{3} - 4040 T^{4} + 83242 T^{5} + 875289 T^{6} - 1023520 T^{7} - 67435611 T^{8} - 1023520 p T^{9} + 875289 p^{2} T^{10} + 83242 p^{3} T^{11} - 4040 p^{4} T^{12} - 1525 p^{5} T^{13} - 52 p^{6} T^{14} + 3 p^{7} T^{15} + p^{8} T^{16} \) |
| 79 | \( 1 + 5 T - 135 T^{2} - 2 p T^{3} + 7850 T^{4} - 39069 T^{5} - 419444 T^{6} + 2703920 T^{7} + 43582455 T^{8} + 2703920 p T^{9} - 419444 p^{2} T^{10} - 39069 p^{3} T^{11} + 7850 p^{4} T^{12} - 2 p^{6} T^{13} - 135 p^{6} T^{14} + 5 p^{7} T^{15} + p^{8} T^{16} \) |
| 83 | \( ( 1 + 12 T + 341 T^{2} + 2700 T^{3} + 42045 T^{4} + 2700 p T^{5} + 341 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 89 | \( 1 + 16 T + 98 T^{2} - 14 p T^{3} - 30047 T^{4} - 328119 T^{5} - 544685 T^{6} + 25067257 T^{7} + 399536363 T^{8} + 25067257 p T^{9} - 544685 p^{2} T^{10} - 328119 p^{3} T^{11} - 30047 p^{4} T^{12} - 14 p^{6} T^{13} + 98 p^{6} T^{14} + 16 p^{7} T^{15} + p^{8} T^{16} \) |
| 97 | \( ( 1 - 3 T + 106 T^{2} - 77 T^{3} + 17313 T^{4} - 77 p T^{5} + 106 p^{2} T^{6} - 3 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
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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
−5.00539478066781348784163397624, −4.76313918271292142923204668836, −4.75631544034136458194263892308, −4.71888797479329130246682177320, −4.50554489988047718856619859156, −4.43464849416804675465302104252, −4.34224482049040034018279990502, −4.05310599555010642140639024516, −3.94649660806114892640753499280, −3.91333640774051036788631337953, −3.80255612589950436935239634392, −3.70767756047626342755291259463, −3.19145637670273737055280134830, −3.05538346314887223498110410369, −2.96203889340790742710002802459, −2.79047753852486087333228231006, −2.51042211698991662965641108304, −2.37966653603208147607110501002, −2.02285385899906717611868777706, −1.65304361823406013325576840998, −1.44321388345946621322416365088, −1.27843446879341671930345780224, −0.985589085149286643107945340895, −0.979365056853101942037530076162, −0.78185345525297621580742082947,
0.78185345525297621580742082947, 0.979365056853101942037530076162, 0.985589085149286643107945340895, 1.27843446879341671930345780224, 1.44321388345946621322416365088, 1.65304361823406013325576840998, 2.02285385899906717611868777706, 2.37966653603208147607110501002, 2.51042211698991662965641108304, 2.79047753852486087333228231006, 2.96203889340790742710002802459, 3.05538346314887223498110410369, 3.19145637670273737055280134830, 3.70767756047626342755291259463, 3.80255612589950436935239634392, 3.91333640774051036788631337953, 3.94649660806114892640753499280, 4.05310599555010642140639024516, 4.34224482049040034018279990502, 4.43464849416804675465302104252, 4.50554489988047718856619859156, 4.71888797479329130246682177320, 4.75631544034136458194263892308, 4.76313918271292142923204668836, 5.00539478066781348784163397624
Plot not available for L-functions of degree greater than 10.
