| L(s) = 1 | − 32·2-s − 14·3-s + 640·4-s + 448·6-s − 1.37e3·7-s − 1.02e4·8-s − 1.54e3·9-s + 9.83e3·11-s − 8.96e3·12-s − 1.77e4·13-s + 4.39e4·14-s + 1.43e5·16-s − 2.91e4·17-s + 4.95e4·18-s − 4.42e3·19-s + 1.92e4·21-s − 3.14e5·22-s − 4.66e4·23-s + 1.43e5·24-s + 5.66e5·26-s + 6.10e4·27-s − 8.78e5·28-s − 1.07e5·29-s + 2.99e5·31-s − 1.83e6·32-s − 1.37e5·33-s + 9.31e5·34-s + ⋯ |
| L(s) = 1 | − 2.82·2-s − 0.299·3-s + 5·4-s + 0.846·6-s − 1.51·7-s − 7.07·8-s − 0.707·9-s + 2.22·11-s − 1.49·12-s − 2.23·13-s + 4.27·14-s + 35/4·16-s − 1.43·17-s + 2.00·18-s − 0.147·19-s + 0.452·21-s − 6.30·22-s − 0.799·23-s + 2.11·24-s + 6.32·26-s + 0.596·27-s − 7.55·28-s − 0.817·29-s + 1.80·31-s − 9.89·32-s − 0.667·33-s + 4.06·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 5^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(8-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 5^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+7/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(4)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{9}{2})\) |
|
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 | $C_1$ | \( ( 1 + p^{3} T )^{4} \) |
| 5 | | \( 1 \) |
| 7 | $C_1$ | \( ( 1 + p^{3} T )^{4} \) |
| good | 3 | $C_2 \wr S_4$ | \( 1 + 14 T + 581 p T^{2} - 4984 p T^{3} + 121772 p^{2} T^{4} - 4984 p^{8} T^{5} + 581 p^{15} T^{6} + 14 p^{21} T^{7} + p^{28} T^{8} \) |
| 11 | $C_2 \wr S_4$ | \( 1 - 9836 T + 92027954 T^{2} - 547370025424 T^{3} + 2836162676652515 T^{4} - 547370025424 p^{7} T^{5} + 92027954 p^{14} T^{6} - 9836 p^{21} T^{7} + p^{28} T^{8} \) |
| 13 | $C_2 \wr S_4$ | \( 1 + 17710 T + 215972852 T^{2} + 1452297069258 T^{3} + 11545519853693222 T^{4} + 1452297069258 p^{7} T^{5} + 215972852 p^{14} T^{6} + 17710 p^{21} T^{7} + p^{28} T^{8} \) |
| 17 | $C_2 \wr S_4$ | \( 1 + 29106 T + 1469115593 T^{2} + 26081106500890 T^{3} + 810716952761799936 T^{4} + 26081106500890 p^{7} T^{5} + 1469115593 p^{14} T^{6} + 29106 p^{21} T^{7} + p^{28} T^{8} \) |
| 19 | $C_2 \wr S_4$ | \( 1 + 4424 T + 1533246583 T^{2} - 8091167463120 T^{3} + 1101162744272825772 T^{4} - 8091167463120 p^{7} T^{5} + 1533246583 p^{14} T^{6} + 4424 p^{21} T^{7} + p^{28} T^{8} \) |
| 23 | $C_2 \wr S_4$ | \( 1 + 46668 T + 9989327803 T^{2} + 278595405247788 T^{3} + 42498057888205754264 T^{4} + 278595405247788 p^{7} T^{5} + 9989327803 p^{14} T^{6} + 46668 p^{21} T^{7} + p^{28} T^{8} \) |
| 29 | $C_2 \wr S_4$ | \( 1 + 107322 T + 34568373649 T^{2} + 1647588470317098 T^{3} + \)\(53\!\cdots\!80\)\( T^{4} + 1647588470317098 p^{7} T^{5} + 34568373649 p^{14} T^{6} + 107322 p^{21} T^{7} + p^{28} T^{8} \) |
| 31 | $C_2 \wr S_4$ | \( 1 - 299810 T + 47730536928 T^{2} - 2730198582027322 T^{3} + \)\(27\!\cdots\!66\)\( T^{4} - 2730198582027322 p^{7} T^{5} + 47730536928 p^{14} T^{6} - 299810 p^{21} T^{7} + p^{28} T^{8} \) |
| 37 | $C_2 \wr S_4$ | \( 1 + 277446 T + 93657906013 T^{2} - 14773065038817350 T^{3} - \)\(52\!\cdots\!44\)\( T^{4} - 14773065038817350 p^{7} T^{5} + 93657906013 p^{14} T^{6} + 277446 p^{21} T^{7} + p^{28} T^{8} \) |
| 41 | $C_2 \wr S_4$ | \( 1 + 76860 T + 316328015513 T^{2} + 8237720043848218 T^{3} + \)\(49\!\cdots\!96\)\( T^{4} + 8237720043848218 p^{7} T^{5} + 316328015513 p^{14} T^{6} + 76860 p^{21} T^{7} + p^{28} T^{8} \) |
| 43 | $C_2 \wr S_4$ | \( 1 + 785316 T + 716947614635 T^{2} + 427626832577394156 T^{3} + \)\(27\!\cdots\!20\)\( T^{4} + 427626832577394156 p^{7} T^{5} + 716947614635 p^{14} T^{6} + 785316 p^{21} T^{7} + p^{28} T^{8} \) |
| 47 | $C_2 \wr S_4$ | \( 1 - 24780 T + 1776391361948 T^{2} + 30882471262300756 T^{3} + \)\(12\!\cdots\!22\)\( T^{4} + 30882471262300756 p^{7} T^{5} + 1776391361948 p^{14} T^{6} - 24780 p^{21} T^{7} + p^{28} T^{8} \) |
| 53 | $C_2 \wr S_4$ | \( 1 + 682896 T + 2284048308480 T^{2} + 1532272007699976336 T^{3} + \)\(30\!\cdots\!50\)\( T^{4} + 1532272007699976336 p^{7} T^{5} + 2284048308480 p^{14} T^{6} + 682896 p^{21} T^{7} + p^{28} T^{8} \) |
| 59 | $C_2 \wr S_4$ | \( 1 + 503846 T + 3493217657728 T^{2} + 1824713630731946710 T^{3} + \)\(14\!\cdots\!62\)\( T^{4} + 1824713630731946710 p^{7} T^{5} + 3493217657728 p^{14} T^{6} + 503846 p^{21} T^{7} + p^{28} T^{8} \) |
| 61 | $C_2 \wr S_4$ | \( 1 - 4527082 T + 19391899154124 T^{2} - 46517081493679543166 T^{3} + \)\(10\!\cdots\!26\)\( T^{4} - 46517081493679543166 p^{7} T^{5} + 19391899154124 p^{14} T^{6} - 4527082 p^{21} T^{7} + p^{28} T^{8} \) |
| 67 | $C_2 \wr S_4$ | \( 1 + 2745156 T + 11593476986218 T^{2} + 31351823725103889840 T^{3} + \)\(11\!\cdots\!11\)\( T^{4} + 31351823725103889840 p^{7} T^{5} + 11593476986218 p^{14} T^{6} + 2745156 p^{21} T^{7} + p^{28} T^{8} \) |
| 71 | $C_2 \wr S_4$ | \( 1 - 7106840 T + 24417539668963 T^{2} - 14945814015180725152 T^{3} - \)\(46\!\cdots\!24\)\( T^{4} - 14945814015180725152 p^{7} T^{5} + 24417539668963 p^{14} T^{6} - 7106840 p^{21} T^{7} + p^{28} T^{8} \) |
| 73 | $C_2 \wr S_4$ | \( 1 - 3542672 T + 44513465758833 T^{2} - \)\(11\!\cdots\!02\)\( T^{3} + \)\(73\!\cdots\!84\)\( T^{4} - \)\(11\!\cdots\!02\)\( p^{7} T^{5} + 44513465758833 p^{14} T^{6} - 3542672 p^{21} T^{7} + p^{28} T^{8} \) |
| 79 | $C_2 \wr S_4$ | \( 1 - 9248908 T + 71210748784759 T^{2} - \)\(41\!\cdots\!92\)\( T^{3} + \)\(21\!\cdots\!60\)\( T^{4} - \)\(41\!\cdots\!92\)\( p^{7} T^{5} + 71210748784759 p^{14} T^{6} - 9248908 p^{21} T^{7} + p^{28} T^{8} \) |
| 83 | $C_2 \wr S_4$ | \( 1 - 2154866 T + 52320946290583 T^{2} - 92604425679962224672 T^{3} + \)\(16\!\cdots\!28\)\( T^{4} - 92604425679962224672 p^{7} T^{5} + 52320946290583 p^{14} T^{6} - 2154866 p^{21} T^{7} + p^{28} T^{8} \) |
| 89 | $C_2 \wr S_4$ | \( 1 - 11328520 T + 133813082540441 T^{2} - \)\(10\!\cdots\!90\)\( T^{3} + \)\(77\!\cdots\!96\)\( T^{4} - \)\(10\!\cdots\!90\)\( p^{7} T^{5} + 133813082540441 p^{14} T^{6} - 11328520 p^{21} T^{7} + p^{28} T^{8} \) |
| 97 | $C_2 \wr S_4$ | \( 1 + 11281774 T + 159012805995052 T^{2} + \)\(17\!\cdots\!86\)\( T^{3} + \)\(43\!\cdots\!14\)\( T^{4} + \)\(17\!\cdots\!86\)\( p^{7} T^{5} + 159012805995052 p^{14} T^{6} + 11281774 p^{21} T^{7} + p^{28} T^{8} \) |
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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
−7.71572883781214282200162655745, −7.32702874317458133974087850185, −7.10870691307371145401034507960, −6.81957758587078072255966366267, −6.69974723165078179649215197329, −6.55488959952750205697079513404, −6.26406111869978925736898724036, −6.13635718131260845480852815172, −6.01589598782126482807663022083, −5.13712255552293936396340525896, −5.10858063998634716643896862019, −5.08443330813825814336081605667, −4.46845964060043640721681460119, −3.99555603064512614127158333020, −3.66877815658178404587382737954, −3.54591261593554977386868506824, −3.33890977218226939482115213022, −2.65308779634426305066427687167, −2.41026780166418895595822473275, −2.36196822108993134919086242238, −2.18379917501354273285742651836, −1.70206790011603504425943910977, −1.17766150646714444947744110077, −1.00465179655113649701234878472, −0.865888377222137100217358947727, 0, 0, 0, 0,
0.865888377222137100217358947727, 1.00465179655113649701234878472, 1.17766150646714444947744110077, 1.70206790011603504425943910977, 2.18379917501354273285742651836, 2.36196822108993134919086242238, 2.41026780166418895595822473275, 2.65308779634426305066427687167, 3.33890977218226939482115213022, 3.54591261593554977386868506824, 3.66877815658178404587382737954, 3.99555603064512614127158333020, 4.46845964060043640721681460119, 5.08443330813825814336081605667, 5.10858063998634716643896862019, 5.13712255552293936396340525896, 6.01589598782126482807663022083, 6.13635718131260845480852815172, 6.26406111869978925736898724036, 6.55488959952750205697079513404, 6.69974723165078179649215197329, 6.81957758587078072255966366267, 7.10870691307371145401034507960, 7.32702874317458133974087850185, 7.71572883781214282200162655745
