L(s) = 1 | + 16·2-s + 64·4-s + 309·5-s + 868·7-s − 1.02e3·8-s + 4.94e3·10-s − 6.74e3·11-s − 1.42e4·13-s + 1.38e4·14-s − 1.63e4·16-s + 2.74e4·17-s + 263·19-s + 1.97e4·20-s − 1.07e5·22-s − 2.75e4·23-s + 1.21e5·25-s − 2.28e5·26-s + 5.55e4·28-s + 7.88e5·29-s + 1.74e5·31-s − 6.55e4·32-s + 4.39e5·34-s + 2.68e5·35-s + 1.06e5·37-s + 4.20e3·38-s − 3.16e5·40-s − 1.13e6·41-s + ⋯ |
L(s) = 1 | + 1.41·2-s + 1/2·4-s + 1.10·5-s + 0.956·7-s − 0.707·8-s + 1.56·10-s − 1.52·11-s − 1.80·13-s + 1.35·14-s − 16-s + 1.35·17-s + 0.00879·19-s + 0.552·20-s − 2.16·22-s − 0.472·23-s + 1.55·25-s − 2.54·26-s + 0.478·28-s + 6.00·29-s + 1.05·31-s − 0.353·32-s + 1.91·34-s + 1.05·35-s + 0.345·37-s + 0.0124·38-s − 0.781·40-s − 2.56·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(8-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+7/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
\(L(4)\) |
\(\approx\) |
\(14.28198768\) |
\(L(\frac12)\) |
\(\approx\) |
\(14.28198768\) |
\(L(\frac{9}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
---|
bad | 2 | $C_2$ | \( ( 1 - p^{3} T + p^{6} T^{2} )^{2} \) |
| 3 | | \( 1 \) |
| 7 | $C_2^2$ | \( 1 - 124 p T + 1647 p^{3} T^{2} - 124 p^{8} T^{3} + p^{14} T^{4} \) |
good | 5 | $D_4\times C_2$ | \( 1 - 309 T - 26299 T^{2} + 2130246 p T^{3} + 50695986 p^{2} T^{4} + 2130246 p^{8} T^{5} - 26299 p^{14} T^{6} - 309 p^{21} T^{7} + p^{28} T^{8} \) |
| 11 | $D_4\times C_2$ | \( 1 + 6747 T - 226613 p T^{2} + 60995646270 T^{3} + 1038571402391724 T^{4} + 60995646270 p^{7} T^{5} - 226613 p^{15} T^{6} + 6747 p^{21} T^{7} + p^{28} T^{8} \) |
| 13 | $D_{4}$ | \( ( 1 + 7139 T + 118629558 T^{2} + 7139 p^{7} T^{3} + p^{14} T^{4} )^{2} \) |
| 17 | $D_4\times C_2$ | \( 1 - 27456 T - 240368290 T^{2} - 4764244193280 T^{3} + 510729866535789219 T^{4} - 4764244193280 p^{7} T^{5} - 240368290 p^{14} T^{6} - 27456 p^{21} T^{7} + p^{28} T^{8} \) |
| 19 | $D_4\times C_2$ | \( 1 - 263 T + 1096190119 T^{2} + 758456344564 T^{3} + 402412623031703488 T^{4} + 758456344564 p^{7} T^{5} + 1096190119 p^{14} T^{6} - 263 p^{21} T^{7} + p^{28} T^{8} \) |
| 23 | $D_4\times C_2$ | \( 1 + 27576 T - 3650157838 T^{2} - 66156403553280 T^{3} + 5271348622384839315 T^{4} - 66156403553280 p^{7} T^{5} - 3650157838 p^{14} T^{6} + 27576 p^{21} T^{7} + p^{28} T^{8} \) |
| 29 | $D_{4}$ | \( ( 1 - 394359 T + 72882131278 T^{2} - 394359 p^{7} T^{3} + p^{14} T^{4} )^{2} \) |
| 31 | $D_4\times C_2$ | \( 1 - 174212 T - 21807641489 T^{2} + 499599213633268 T^{3} + \)\(11\!\cdots\!84\)\( T^{4} + 499599213633268 p^{7} T^{5} - 21807641489 p^{14} T^{6} - 174212 p^{21} T^{7} + p^{28} T^{8} \) |
| 37 | $D_4\times C_2$ | \( 1 - 106589 T - 56814354305 T^{2} + 12970621955228560 T^{3} - \)\(54\!\cdots\!06\)\( T^{4} + 12970621955228560 p^{7} T^{5} - 56814354305 p^{14} T^{6} - 106589 p^{21} T^{7} + p^{28} T^{8} \) |
| 41 | $D_{4}$ | \( ( 1 + 567006 T + 469574436322 T^{2} + 567006 p^{7} T^{3} + p^{14} T^{4} )^{2} \) |
| 43 | $D_{4}$ | \( ( 1 + 265505 T - 24378589836 T^{2} + 265505 p^{7} T^{3} + p^{14} T^{4} )^{2} \) |
| 47 | $D_4\times C_2$ | \( 1 - 520092 T - 786677028034 T^{2} - 22845808533192624 T^{3} + \)\(72\!\cdots\!03\)\( T^{4} - 22845808533192624 p^{7} T^{5} - 786677028034 p^{14} T^{6} - 520092 p^{21} T^{7} + p^{28} T^{8} \) |
| 53 | $D_4\times C_2$ | \( 1 + 366747 T - 1966776590437 T^{2} - 91005454083887316 T^{3} + \)\(28\!\cdots\!14\)\( T^{4} - 91005454083887316 p^{7} T^{5} - 1966776590437 p^{14} T^{6} + 366747 p^{21} T^{7} + p^{28} T^{8} \) |
| 59 | $D_4\times C_2$ | \( 1 - 1131963 T - 3893468173921 T^{2} - 223568847720870876 T^{3} + \)\(17\!\cdots\!28\)\( T^{4} - 223568847720870876 p^{7} T^{5} - 3893468173921 p^{14} T^{6} - 1131963 p^{21} T^{7} + p^{28} T^{8} \) |
| 61 | $D_4\times C_2$ | \( 1 + 614488 T - 4376677546838 T^{2} - 940911782318893280 T^{3} + \)\(11\!\cdots\!59\)\( T^{4} - 940911782318893280 p^{7} T^{5} - 4376677546838 p^{14} T^{6} + 614488 p^{21} T^{7} + p^{28} T^{8} \) |
| 67 | $D_4\times C_2$ | \( 1 - 4777883 T + 5495885093353 T^{2} - 24896868218096870270 T^{3} + \)\(12\!\cdots\!20\)\( T^{4} - 24896868218096870270 p^{7} T^{5} + 5495885093353 p^{14} T^{6} - 4777883 p^{21} T^{7} + p^{28} T^{8} \) |
| 71 | $D_{4}$ | \( ( 1 - 93468 p T + 26504583065314 T^{2} - 93468 p^{8} T^{3} + p^{14} T^{4} )^{2} \) |
| 73 | $D_4\times C_2$ | \( 1 + 4647343 T + 4067993097973 T^{2} - 21215089083938691674 T^{3} - \)\(53\!\cdots\!86\)\( T^{4} - 21215089083938691674 p^{7} T^{5} + 4067993097973 p^{14} T^{6} + 4647343 p^{21} T^{7} + p^{28} T^{8} \) |
| 79 | $D_4\times C_2$ | \( 1 - 1334750 T - 29702831180087 T^{2} + 9241047164383452250 T^{3} + \)\(58\!\cdots\!88\)\( T^{4} + 9241047164383452250 p^{7} T^{5} - 29702831180087 p^{14} T^{6} - 1334750 p^{21} T^{7} + p^{28} T^{8} \) |
| 83 | $D_{4}$ | \( ( 1 - 1217247 T + 40950523694344 T^{2} - 1217247 p^{7} T^{3} + p^{14} T^{4} )^{2} \) |
| 89 | $D_4\times C_2$ | \( 1 + 13466826 T + 49099609377938 T^{2} + \)\(58\!\cdots\!80\)\( T^{3} + \)\(75\!\cdots\!99\)\( T^{4} + \)\(58\!\cdots\!80\)\( p^{7} T^{5} + 49099609377938 p^{14} T^{6} + 13466826 p^{21} T^{7} + p^{28} T^{8} \) |
| 97 | $D_{4}$ | \( ( 1 + 16705031 T + 230323942277784 T^{2} + 16705031 p^{7} T^{3} + p^{14} T^{4} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.408707569321783138869031537387, −8.173393009283885354565689161952, −8.000179092849740918831188015569, −7.62142787597911792421564823366, −6.97066094360976925550607901502, −6.89642499273106019648211306524, −6.64426678921969472717213084854, −6.22743559080861948272254628126, −6.11885951929670038512891650077, −5.39098452166741757922790565759, −5.20430460830833583181992247633, −5.07349193482214287367501818551, −4.98369266011780231581291060544, −4.59587585013352911739962395089, −4.43333225318233157941966788912, −3.82230550885388889696934312318, −3.22414169525130636842669352230, −3.06574014024984621132339831846, −2.56348558241591283859456344450, −2.52955518284478795245535218501, −2.24521132463331394526493985901, −1.46321373441679466111755845585, −1.14089550235967894811620031520, −0.70604931380583232197881561017, −0.39609920442371930301951866582,
0.39609920442371930301951866582, 0.70604931380583232197881561017, 1.14089550235967894811620031520, 1.46321373441679466111755845585, 2.24521132463331394526493985901, 2.52955518284478795245535218501, 2.56348558241591283859456344450, 3.06574014024984621132339831846, 3.22414169525130636842669352230, 3.82230550885388889696934312318, 4.43333225318233157941966788912, 4.59587585013352911739962395089, 4.98369266011780231581291060544, 5.07349193482214287367501818551, 5.20430460830833583181992247633, 5.39098452166741757922790565759, 6.11885951929670038512891650077, 6.22743559080861948272254628126, 6.64426678921969472717213084854, 6.89642499273106019648211306524, 6.97066094360976925550607901502, 7.62142787597911792421564823366, 8.000179092849740918831188015569, 8.173393009283885354565689161952, 8.408707569321783138869031537387