| L(s) = 1 | − 32·2-s + 21·3-s + 640·4-s − 735·5-s − 672·6-s − 1.37e3·7-s − 1.02e4·8-s − 3.09e3·9-s + 2.35e4·10-s + 5.32e3·11-s + 1.34e4·12-s + 9.17e3·13-s + 4.39e4·14-s − 1.54e4·15-s + 1.43e5·16-s + 4.98e3·17-s + 9.91e4·18-s − 6.11e3·19-s − 4.70e5·20-s − 2.88e4·21-s − 1.70e5·22-s − 2.36e4·23-s − 2.15e5·24-s + 1.07e5·25-s − 2.93e5·26-s − 6.86e4·27-s − 8.78e5·28-s + ⋯ |
| L(s) = 1 | − 2.82·2-s + 0.449·3-s + 5·4-s − 2.62·5-s − 1.27·6-s − 1.51·7-s − 7.07·8-s − 1.41·9-s + 7.43·10-s + 1.20·11-s + 2.24·12-s + 1.15·13-s + 4.27·14-s − 1.18·15-s + 35/4·16-s + 0.246·17-s + 4.00·18-s − 0.204·19-s − 13.1·20-s − 0.678·21-s − 3.41·22-s − 0.404·23-s − 3.17·24-s + 1.37·25-s − 3.27·26-s − 0.671·27-s − 7.55·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 7^{4} \cdot 11^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(8-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 7^{4} \cdot 11^{4}\right)^{s/2} \, \Gamma_{\C}(s+7/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(4)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{9}{2})\) |
|
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 | $C_1$ | \( ( 1 + p^{3} T )^{4} \) |
| 7 | $C_1$ | \( ( 1 + p^{3} T )^{4} \) |
| 11 | $C_1$ | \( ( 1 - p^{3} T )^{4} \) |
| good | 3 | $C_2 \wr C_2\wr C_2$ | \( 1 - 7 p T + 1180 p T^{2} - 7861 p^{2} T^{3} + 456346 p^{3} T^{4} - 7861 p^{9} T^{5} + 1180 p^{15} T^{6} - 7 p^{22} T^{7} + p^{28} T^{8} \) |
| 5 | $C_2 \wr C_2\wr C_2$ | \( 1 + 147 p T + 17298 p^{2} T^{2} + 285789 p^{4} T^{3} + 88722458 p^{4} T^{4} + 285789 p^{11} T^{5} + 17298 p^{16} T^{6} + 147 p^{22} T^{7} + p^{28} T^{8} \) |
| 13 | $C_2 \wr C_2\wr C_2$ | \( 1 - 9170 T + 141138664 T^{2} - 1405263952526 T^{3} + 11454425351213470 T^{4} - 1405263952526 p^{7} T^{5} + 141138664 p^{14} T^{6} - 9170 p^{21} T^{7} + p^{28} T^{8} \) |
| 17 | $C_2 \wr C_2\wr C_2$ | \( 1 - 4984 T + 1054478148 T^{2} - 894646925760 T^{3} + 549430287818102566 T^{4} - 894646925760 p^{7} T^{5} + 1054478148 p^{14} T^{6} - 4984 p^{21} T^{7} + p^{28} T^{8} \) |
| 19 | $C_2 \wr C_2\wr C_2$ | \( 1 + 322 p T + 1068206876 T^{2} - 5108706599514 T^{3} + 14223420780044594 p T^{4} - 5108706599514 p^{7} T^{5} + 1068206876 p^{14} T^{6} + 322 p^{22} T^{7} + p^{28} T^{8} \) |
| 23 | $C_2 \wr C_2\wr C_2$ | \( 1 + 23601 T + 8113806708 T^{2} + 52922846965317 T^{3} + 33056002682315873078 T^{4} + 52922846965317 p^{7} T^{5} + 8113806708 p^{14} T^{6} + 23601 p^{21} T^{7} + p^{28} T^{8} \) |
| 29 | $C_2 \wr C_2\wr C_2$ | \( 1 - 5156 p T + 35469773636 T^{2} - 2241034856188 p^{2} T^{3} + \)\(47\!\cdots\!86\)\( T^{4} - 2241034856188 p^{9} T^{5} + 35469773636 p^{14} T^{6} - 5156 p^{22} T^{7} + p^{28} T^{8} \) |
| 31 | $C_2 \wr C_2\wr C_2$ | \( 1 + 13167 T + 89123081468 T^{2} + 2481034352348979 T^{3} + \)\(33\!\cdots\!70\)\( T^{4} + 2481034352348979 p^{7} T^{5} + 89123081468 p^{14} T^{6} + 13167 p^{21} T^{7} + p^{28} T^{8} \) |
| 37 | $C_2 \wr C_2\wr C_2$ | \( 1 - 200309 T + 348411752318 T^{2} - 51997568644629615 T^{3} + \)\(48\!\cdots\!06\)\( T^{4} - 51997568644629615 p^{7} T^{5} + 348411752318 p^{14} T^{6} - 200309 p^{21} T^{7} + p^{28} T^{8} \) |
| 41 | $C_2 \wr C_2\wr C_2$ | \( 1 - 134876 T + 600492075756 T^{2} - 60817391484845436 T^{3} + \)\(16\!\cdots\!50\)\( T^{4} - 60817391484845436 p^{7} T^{5} + 600492075756 p^{14} T^{6} - 134876 p^{21} T^{7} + p^{28} T^{8} \) |
| 43 | $C_2 \wr C_2\wr C_2$ | \( 1 - 1024644 T + 1066135967948 T^{2} - 631210007692837668 T^{3} + \)\(41\!\cdots\!18\)\( T^{4} - 631210007692837668 p^{7} T^{5} + 1066135967948 p^{14} T^{6} - 1024644 p^{21} T^{7} + p^{28} T^{8} \) |
| 47 | $C_2 \wr C_2\wr C_2$ | \( 1 + 153328 T + 1319159955572 T^{2} + 108400262744758264 T^{3} + \)\(82\!\cdots\!38\)\( T^{4} + 108400262744758264 p^{7} T^{5} + 1319159955572 p^{14} T^{6} + 153328 p^{21} T^{7} + p^{28} T^{8} \) |
| 53 | $C_2 \wr C_2\wr C_2$ | \( 1 + 2464172 T + 6775725938052 T^{2} + 9341425244431170180 T^{3} + \)\(13\!\cdots\!26\)\( T^{4} + 9341425244431170180 p^{7} T^{5} + 6775725938052 p^{14} T^{6} + 2464172 p^{21} T^{7} + p^{28} T^{8} \) |
| 59 | $C_2 \wr C_2\wr C_2$ | \( 1 + 4604453 T + 10617223290636 T^{2} + 18734745360509020101 T^{3} + \)\(30\!\cdots\!46\)\( T^{4} + 18734745360509020101 p^{7} T^{5} + 10617223290636 p^{14} T^{6} + 4604453 p^{21} T^{7} + p^{28} T^{8} \) |
| 61 | $C_2 \wr C_2\wr C_2$ | \( 1 + 2750468 T + 7429384516132 T^{2} + 15138283361482406804 T^{3} + \)\(35\!\cdots\!70\)\( T^{4} + 15138283361482406804 p^{7} T^{5} + 7429384516132 p^{14} T^{6} + 2750468 p^{21} T^{7} + p^{28} T^{8} \) |
| 67 | $C_2 \wr C_2\wr C_2$ | \( 1 + 3083243 T + 13058810026472 T^{2} + 4766337135738099579 T^{3} + \)\(35\!\cdots\!18\)\( T^{4} + 4766337135738099579 p^{7} T^{5} + 13058810026472 p^{14} T^{6} + 3083243 p^{21} T^{7} + p^{28} T^{8} \) |
| 71 | $C_2 \wr C_2\wr C_2$ | \( 1 + 1175531 T + 18041767470164 T^{2} + 2388275421623095319 T^{3} + \)\(15\!\cdots\!10\)\( T^{4} + 2388275421623095319 p^{7} T^{5} + 18041767470164 p^{14} T^{6} + 1175531 p^{21} T^{7} + p^{28} T^{8} \) |
| 73 | $C_2 \wr C_2\wr C_2$ | \( 1 + 3730328 T + 6133896480916 T^{2} + 32984152761895656608 T^{3} + \)\(22\!\cdots\!14\)\( T^{4} + 32984152761895656608 p^{7} T^{5} + 6133896480916 p^{14} T^{6} + 3730328 p^{21} T^{7} + p^{28} T^{8} \) |
| 79 | $C_2 \wr C_2\wr C_2$ | \( 1 + 13248130 T + 89188608357836 T^{2} + \)\(36\!\cdots\!10\)\( T^{3} + \)\(14\!\cdots\!86\)\( T^{4} + \)\(36\!\cdots\!10\)\( p^{7} T^{5} + 89188608357836 p^{14} T^{6} + 13248130 p^{21} T^{7} + p^{28} T^{8} \) |
| 83 | $C_2 \wr C_2\wr C_2$ | \( 1 + 10677548 T + 82391079851676 T^{2} + \)\(38\!\cdots\!08\)\( T^{3} + \)\(20\!\cdots\!34\)\( T^{4} + \)\(38\!\cdots\!08\)\( p^{7} T^{5} + 82391079851676 p^{14} T^{6} + 10677548 p^{21} T^{7} + p^{28} T^{8} \) |
| 89 | $C_2 \wr C_2\wr C_2$ | \( 1 + 20682725 T + 244094449448166 T^{2} + \)\(19\!\cdots\!75\)\( T^{3} + \)\(13\!\cdots\!46\)\( T^{4} + \)\(19\!\cdots\!75\)\( p^{7} T^{5} + 244094449448166 p^{14} T^{6} + 20682725 p^{21} T^{7} + p^{28} T^{8} \) |
| 97 | $C_2 \wr C_2\wr C_2$ | \( 1 + 4902835 T - 28635347633614 T^{2} + \)\(42\!\cdots\!53\)\( T^{3} + \)\(12\!\cdots\!90\)\( T^{4} + \)\(42\!\cdots\!53\)\( p^{7} T^{5} - 28635347633614 p^{14} T^{6} + 4902835 p^{21} T^{7} + p^{28} T^{8} \) |
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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
−8.704289292353324831418893530748, −8.221916451007085323361659375215, −8.155290895353517161290452286089, −8.108786605789618150049084827499, −7.896282420035265708064130924300, −7.30974466252074457803225648579, −7.27828423218582355885783233523, −6.92340337098620568468752681744, −6.82633552574737237727325167386, −5.99015232209730009452808377218, −5.95432610736154224061847955745, −5.95336876831846766532878520839, −5.88404017592929143721221844360, −4.67982163929780492658887633572, −4.25715302317711046537756556330, −4.02184608304950227603337767661, −3.97477669526772834105510950131, −3.22093991622098388632086694897, −2.96843754232797443866739371637, −2.89977389118619677814894931243, −2.86044440207440425680689464767, −1.75297595855204879837084429392, −1.66753216520367727564122134533, −1.06655507475682119843061816942, −0.956482452413325975630330051160, 0, 0, 0, 0,
0.956482452413325975630330051160, 1.06655507475682119843061816942, 1.66753216520367727564122134533, 1.75297595855204879837084429392, 2.86044440207440425680689464767, 2.89977389118619677814894931243, 2.96843754232797443866739371637, 3.22093991622098388632086694897, 3.97477669526772834105510950131, 4.02184608304950227603337767661, 4.25715302317711046537756556330, 4.67982163929780492658887633572, 5.88404017592929143721221844360, 5.95336876831846766532878520839, 5.95432610736154224061847955745, 5.99015232209730009452808377218, 6.82633552574737237727325167386, 6.92340337098620568468752681744, 7.27828423218582355885783233523, 7.30974466252074457803225648579, 7.896282420035265708064130924300, 8.108786605789618150049084827499, 8.155290895353517161290452286089, 8.221916451007085323361659375215, 8.704289292353324831418893530748
