| L(s) = 1 | − 35·3-s + 46·4-s + 537·5-s + 170·7-s − 140·8-s − 2.85e3·9-s − 5.32e3·11-s − 1.61e3·12-s + 4.25e3·13-s − 1.87e4·15-s − 5.08e3·16-s + 5.43e4·17-s + 6.78e4·19-s + 2.47e4·20-s − 5.95e3·21-s − 9.01e3·23-s + 4.90e3·24-s − 800·25-s + 9.85e4·27-s + 7.82e3·28-s + 2.34e5·29-s + 1.89e5·31-s − 3.08e4·32-s + 1.86e5·33-s + 9.12e4·35-s − 1.31e5·36-s + 1.27e5·37-s + ⋯ |
| L(s) = 1 | − 0.748·3-s + 0.359·4-s + 1.92·5-s + 0.187·7-s − 0.0966·8-s − 1.30·9-s − 1.20·11-s − 0.268·12-s + 0.536·13-s − 1.43·15-s − 0.310·16-s + 2.68·17-s + 2.26·19-s + 0.690·20-s − 0.140·21-s − 0.154·23-s + 0.0723·24-s − 0.0102·25-s + 0.963·27-s + 0.0673·28-s + 1.78·29-s + 1.14·31-s − 0.166·32-s + 0.902·33-s + 0.359·35-s − 0.468·36-s + 0.415·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 14641 ^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 14641 ^{s/2} \, \Gamma_{\C}(s+7/2)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(2.736219833\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.736219833\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 11 | $C_1$ | \( ( 1 + p^{3} T )^{4} \) |
| good | 2 | $C_2 \wr S_4$ | \( 1 - 23 p T^{2} + 35 p^{2} T^{3} + 225 p^{5} T^{4} + 35 p^{9} T^{5} - 23 p^{15} T^{6} + p^{28} T^{8} \) |
| 3 | $C_2 \wr S_4$ | \( 1 + 35 T + 4075 T^{2} + 15980 p^{2} T^{3} + 1103564 p^{2} T^{4} + 15980 p^{9} T^{5} + 4075 p^{14} T^{6} + 35 p^{21} T^{7} + p^{28} T^{8} \) |
| 5 | $C_2 \wr S_4$ | \( 1 - 537 T + 289169 T^{2} - 21850106 p T^{3} + 1357177734 p^{2} T^{4} - 21850106 p^{8} T^{5} + 289169 p^{14} T^{6} - 537 p^{21} T^{7} + p^{28} T^{8} \) |
| 7 | $C_2 \wr S_4$ | \( 1 - 170 T + 227368 p T^{2} + 385746230 T^{3} + 1352587842542 T^{4} + 385746230 p^{7} T^{5} + 227368 p^{15} T^{6} - 170 p^{21} T^{7} + p^{28} T^{8} \) |
| 13 | $C_2 \wr S_4$ | \( 1 - 4250 T + 146352268 T^{2} - 163565755750 T^{3} + 9973548692894534 T^{4} - 163565755750 p^{7} T^{5} + 146352268 p^{14} T^{6} - 4250 p^{21} T^{7} + p^{28} T^{8} \) |
| 17 | $C_2 \wr S_4$ | \( 1 - 54300 T + 2339944100 T^{2} - 68617141197220 T^{3} + 1577703633559808886 T^{4} - 68617141197220 p^{7} T^{5} + 2339944100 p^{14} T^{6} - 54300 p^{21} T^{7} + p^{28} T^{8} \) |
| 19 | $C_2 \wr S_4$ | \( 1 - 67844 T + 3988659532 T^{2} - 151149892040772 T^{3} + 5214797241291386070 T^{4} - 151149892040772 p^{7} T^{5} + 3988659532 p^{14} T^{6} - 67844 p^{21} T^{7} + p^{28} T^{8} \) |
| 23 | $C_2 \wr S_4$ | \( 1 + 9015 T + 5859983975 T^{2} + 1114173371080 T^{3} + 27480155779765016976 T^{4} + 1114173371080 p^{7} T^{5} + 5859983975 p^{14} T^{6} + 9015 p^{21} T^{7} + p^{28} T^{8} \) |
| 29 | $C_2 \wr S_4$ | \( 1 - 234078 T + 45943187444 T^{2} - 4197289348403874 T^{3} + \)\(57\!\cdots\!66\)\( T^{4} - 4197289348403874 p^{7} T^{5} + 45943187444 p^{14} T^{6} - 234078 p^{21} T^{7} + p^{28} T^{8} \) |
| 31 | $C_2 \wr S_4$ | \( 1 - 189857 T + 109086552559 T^{2} - 15043423507120000 T^{3} + \)\(44\!\cdots\!20\)\( T^{4} - 15043423507120000 p^{7} T^{5} + 109086552559 p^{14} T^{6} - 189857 p^{21} T^{7} + p^{28} T^{8} \) |
| 37 | $C_2 \wr S_4$ | \( 1 - 127895 T + 134838507565 T^{2} - 48759217149453270 T^{3} + \)\(11\!\cdots\!06\)\( T^{4} - 48759217149453270 p^{7} T^{5} + 134838507565 p^{14} T^{6} - 127895 p^{21} T^{7} + p^{28} T^{8} \) |
| 41 | $C_2 \wr S_4$ | \( 1 - 289842 T + 612672453332 T^{2} - 98108268285963238 T^{3} + \)\(15\!\cdots\!78\)\( T^{4} - 98108268285963238 p^{7} T^{5} + 612672453332 p^{14} T^{6} - 289842 p^{21} T^{7} + p^{28} T^{8} \) |
| 43 | $C_2 \wr S_4$ | \( 1 - 704930 T + 844898979712 T^{2} - 468056840641753290 T^{3} + \)\(31\!\cdots\!34\)\( T^{4} - 468056840641753290 p^{7} T^{5} + 844898979712 p^{14} T^{6} - 704930 p^{21} T^{7} + p^{28} T^{8} \) |
| 47 | $C_2 \wr S_4$ | \( 1 + 1729080 T + 2943241041980 T^{2} + 2755356691646036440 T^{3} + \)\(24\!\cdots\!38\)\( T^{4} + 2755356691646036440 p^{7} T^{5} + 2943241041980 p^{14} T^{6} + 1729080 p^{21} T^{7} + p^{28} T^{8} \) |
| 53 | $C_2 \wr S_4$ | \( 1 - 1098660 T + 3684975589940 T^{2} - 2958455245352196780 T^{3} + \)\(60\!\cdots\!06\)\( T^{4} - 2958455245352196780 p^{7} T^{5} + 3684975589940 p^{14} T^{6} - 1098660 p^{21} T^{7} + p^{28} T^{8} \) |
| 59 | $C_2 \wr S_4$ | \( 1 + 4665777 T + 13811331261011 T^{2} + 31029840262037202148 T^{3} + \)\(56\!\cdots\!60\)\( T^{4} + 31029840262037202148 p^{7} T^{5} + 13811331261011 p^{14} T^{6} + 4665777 p^{21} T^{7} + p^{28} T^{8} \) |
| 61 | $C_2 \wr S_4$ | \( 1 - 310610 T + 10401137851228 T^{2} - 2935186858120476750 T^{3} + \)\(45\!\cdots\!78\)\( T^{4} - 2935186858120476750 p^{7} T^{5} + 10401137851228 p^{14} T^{6} - 310610 p^{21} T^{7} + p^{28} T^{8} \) |
| 67 | $C_2 \wr S_4$ | \( 1 + 3368245 T + 391116105937 p T^{2} + 59223763433080251700 T^{3} + \)\(24\!\cdots\!12\)\( T^{4} + 59223763433080251700 p^{7} T^{5} + 391116105937 p^{15} T^{6} + 3368245 p^{21} T^{7} + p^{28} T^{8} \) |
| 71 | $C_2 \wr S_4$ | \( 1 + 3416541 T + 19147459152479 T^{2} + 50840343388526872176 T^{3} + \)\(24\!\cdots\!60\)\( T^{4} + 50840343388526872176 p^{7} T^{5} + 19147459152479 p^{14} T^{6} + 3416541 p^{21} T^{7} + p^{28} T^{8} \) |
| 73 | $C_2 \wr S_4$ | \( 1 - 11466230 T + 83249782918924 T^{2} - \)\(40\!\cdots\!70\)\( T^{3} + \)\(15\!\cdots\!62\)\( T^{4} - \)\(40\!\cdots\!70\)\( p^{7} T^{5} + 83249782918924 p^{14} T^{6} - 11466230 p^{21} T^{7} + p^{28} T^{8} \) |
| 79 | $C_2 \wr S_4$ | \( 1 - 566282 T + 54686769020584 T^{2} - 16962398190802371386 T^{3} + \)\(13\!\cdots\!66\)\( T^{4} - 16962398190802371386 p^{7} T^{5} + 54686769020584 p^{14} T^{6} - 566282 p^{21} T^{7} + p^{28} T^{8} \) |
| 83 | $C_2 \wr S_4$ | \( 1 + 4220790 T + 53047302130736 T^{2} + \)\(18\!\cdots\!50\)\( T^{3} + \)\(21\!\cdots\!82\)\( T^{4} + \)\(18\!\cdots\!50\)\( p^{7} T^{5} + 53047302130736 p^{14} T^{6} + 4220790 p^{21} T^{7} + p^{28} T^{8} \) |
| 89 | $C_2 \wr S_4$ | \( 1 - 18265191 T + 262942921135913 T^{2} - \)\(24\!\cdots\!06\)\( T^{3} + \)\(19\!\cdots\!38\)\( T^{4} - \)\(24\!\cdots\!06\)\( p^{7} T^{5} + 262942921135913 p^{14} T^{6} - 18265191 p^{21} T^{7} + p^{28} T^{8} \) |
| 97 | $C_2 \wr S_4$ | \( 1 - 11425325 T + 204974824565293 T^{2} - \)\(18\!\cdots\!30\)\( T^{3} + \)\(24\!\cdots\!26\)\( T^{4} - \)\(18\!\cdots\!30\)\( p^{7} T^{5} + 204974824565293 p^{14} T^{6} - 11425325 p^{21} T^{7} + p^{28} T^{8} \) |
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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
−14.28500223929598701703553480126, −13.53032023002103601774633084316, −13.52363938724005438363562838469, −13.10122524915071680121589525414, −12.06560535172230628117222964736, −12.05942637251109490579359737576, −11.92462333409280725515677672997, −11.12291425256694568507851795797, −10.87100134357732637039080726975, −10.27986800613411480287336390760, −9.911744712600821316846616104027, −9.475435367137605336010053535864, −9.395896976432149188151278672353, −8.255709925296289019289045455423, −7.79755299562503315156106280841, −7.78728569724580109017417912197, −6.47457005282767244384449341095, −6.15363547885793939753041524206, −5.58153808374209993629342166147, −5.50343168201262409154254978575, −4.87734968517136395285072199095, −3.09199739069876330913918495664, −2.94550634386887085371943868282, −1.70499658332364073832658721927, −0.78620566004668121914165103995,
0.78620566004668121914165103995, 1.70499658332364073832658721927, 2.94550634386887085371943868282, 3.09199739069876330913918495664, 4.87734968517136395285072199095, 5.50343168201262409154254978575, 5.58153808374209993629342166147, 6.15363547885793939753041524206, 6.47457005282767244384449341095, 7.78728569724580109017417912197, 7.79755299562503315156106280841, 8.255709925296289019289045455423, 9.395896976432149188151278672353, 9.475435367137605336010053535864, 9.911744712600821316846616104027, 10.27986800613411480287336390760, 10.87100134357732637039080726975, 11.12291425256694568507851795797, 11.92462333409280725515677672997, 12.05942637251109490579359737576, 12.06560535172230628117222964736, 13.10122524915071680121589525414, 13.52363938724005438363562838469, 13.53032023002103601774633084316, 14.28500223929598701703553480126
