| L(s) = 1 | − 32·2-s − 108·3-s + 640·4-s − 134·5-s + 3.45e3·6-s + 1.37e3·7-s − 1.02e4·8-s + 7.29e3·9-s + 4.28e3·10-s − 5.44e3·11-s − 6.91e4·12-s − 8.78e3·13-s − 4.39e4·14-s + 1.44e4·15-s + 1.43e5·16-s + 4.54e4·17-s − 2.33e5·18-s − 1.11e4·19-s − 8.57e4·20-s − 1.48e5·21-s + 1.74e5·22-s + 1.77e4·23-s + 1.10e6·24-s − 1.14e5·25-s + 2.81e5·26-s − 3.93e5·27-s + 8.78e5·28-s + ⋯ |
| L(s) = 1 | − 2.82·2-s − 2.30·3-s + 5·4-s − 0.479·5-s + 6.53·6-s + 1.51·7-s − 7.07·8-s + 10/3·9-s + 1.35·10-s − 1.23·11-s − 11.5·12-s − 1.10·13-s − 4.27·14-s + 1.10·15-s + 35/4·16-s + 2.24·17-s − 9.42·18-s − 0.374·19-s − 2.39·20-s − 3.49·21-s + 3.48·22-s + 0.304·23-s + 16.3·24-s − 1.46·25-s + 3.13·26-s − 3.84·27-s + 7.55·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 7^{4} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(8-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 7^{4} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s+7/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
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 | $C_1$ | \( ( 1 + p^{3} T )^{4} \) |
| 3 | $C_1$ | \( ( 1 + p^{3} T )^{4} \) |
| 7 | $C_1$ | \( ( 1 - p^{3} T )^{4} \) |
| 13 | $C_1$ | \( ( 1 + p^{3} T )^{4} \) |
| good | 5 | $C_2 \wr S_4$ | \( 1 + 134 T + 26553 p T^{2} + 131802 p^{2} T^{3} + 94472288 p^{3} T^{4} + 131802 p^{9} T^{5} + 26553 p^{15} T^{6} + 134 p^{21} T^{7} + p^{28} T^{8} \) |
| 11 | $C_2 \wr S_4$ | \( 1 + 5443 T + 65365430 T^{2} + 282704165119 T^{3} + 1810939996241242 T^{4} + 282704165119 p^{7} T^{5} + 65365430 p^{14} T^{6} + 5443 p^{21} T^{7} + p^{28} T^{8} \) |
| 17 | $C_2 \wr S_4$ | \( 1 - 45437 T + 1891844196 T^{2} - 52810957376307 T^{3} + 1221335472326321830 T^{4} - 52810957376307 p^{7} T^{5} + 1891844196 p^{14} T^{6} - 45437 p^{21} T^{7} + p^{28} T^{8} \) |
| 19 | $C_2 \wr S_4$ | \( 1 + 11194 T + 2411350177 T^{2} + 35276294875594 T^{3} + 2755563998553419620 T^{4} + 35276294875594 p^{7} T^{5} + 2411350177 p^{14} T^{6} + 11194 p^{21} T^{7} + p^{28} T^{8} \) |
| 23 | $C_2 \wr S_4$ | \( 1 - 17762 T + 11387188737 T^{2} - 137887101122778 T^{3} + 54737803760871819220 T^{4} - 137887101122778 p^{7} T^{5} + 11387188737 p^{14} T^{6} - 17762 p^{21} T^{7} + p^{28} T^{8} \) |
| 29 | $C_2 \wr S_4$ | \( 1 + 24980 T + 38797907751 T^{2} + 1687585700610192 T^{3} + \)\(86\!\cdots\!84\)\( T^{4} + 1687585700610192 p^{7} T^{5} + 38797907751 p^{14} T^{6} + 24980 p^{21} T^{7} + p^{28} T^{8} \) |
| 31 | $C_2 \wr S_4$ | \( 1 + 15419 T + 66523692256 T^{2} + 278966779672991 T^{3} + \)\(22\!\cdots\!26\)\( T^{4} + 278966779672991 p^{7} T^{5} + 66523692256 p^{14} T^{6} + 15419 p^{21} T^{7} + p^{28} T^{8} \) |
| 37 | $C_2 \wr S_4$ | \( 1 + 602081 T + 399970492132 T^{2} + 157827619525865531 T^{3} + \)\(56\!\cdots\!66\)\( T^{4} + 157827619525865531 p^{7} T^{5} + 399970492132 p^{14} T^{6} + 602081 p^{21} T^{7} + p^{28} T^{8} \) |
| 41 | $C_2 \wr S_4$ | \( 1 + 596814 T + 557054621060 T^{2} + 175118972689251738 T^{3} + \)\(11\!\cdots\!06\)\( T^{4} + 175118972689251738 p^{7} T^{5} + 557054621060 p^{14} T^{6} + 596814 p^{21} T^{7} + p^{28} T^{8} \) |
| 43 | $C_2 \wr S_4$ | \( 1 + 697726 T + 1140749015249 T^{2} + 528474955739964390 T^{3} + \)\(46\!\cdots\!36\)\( T^{4} + 528474955739964390 p^{7} T^{5} + 1140749015249 p^{14} T^{6} + 697726 p^{21} T^{7} + p^{28} T^{8} \) |
| 47 | $C_2 \wr S_4$ | \( 1 + 238097 T + 1742214469148 T^{2} + 263571600159260633 T^{3} + \)\(26\!\cdots\!30\)\( p T^{4} + 263571600159260633 p^{7} T^{5} + 1742214469148 p^{14} T^{6} + 238097 p^{21} T^{7} + p^{28} T^{8} \) |
| 53 | $C_2 \wr S_4$ | \( 1 - 752377 T + 3890166003878 T^{2} - 1929475750738686667 T^{3} + \)\(62\!\cdots\!06\)\( T^{4} - 1929475750738686667 p^{7} T^{5} + 3890166003878 p^{14} T^{6} - 752377 p^{21} T^{7} + p^{28} T^{8} \) |
| 59 | $C_2 \wr S_4$ | \( 1 - 1651268 T + 5321466533976 T^{2} - 7025396497735275204 T^{3} + \)\(12\!\cdots\!66\)\( T^{4} - 7025396497735275204 p^{7} T^{5} + 5321466533976 p^{14} T^{6} - 1651268 p^{21} T^{7} + p^{28} T^{8} \) |
| 61 | $C_2 \wr S_4$ | \( 1 - 178667 T + 7591355995922 T^{2} - 3282126913980361833 T^{3} + \)\(29\!\cdots\!02\)\( T^{4} - 3282126913980361833 p^{7} T^{5} + 7591355995922 p^{14} T^{6} - 178667 p^{21} T^{7} + p^{28} T^{8} \) |
| 67 | $C_2 \wr S_4$ | \( 1 - 658872 T + 5545736172536 T^{2} - 1078030583545639512 T^{3} + \)\(60\!\cdots\!70\)\( T^{4} - 1078030583545639512 p^{7} T^{5} + 5545736172536 p^{14} T^{6} - 658872 p^{21} T^{7} + p^{28} T^{8} \) |
| 71 | $C_2 \wr S_4$ | \( 1 - 191640 T + 151767139312 p T^{2} + 7908087460092535224 T^{3} + \)\(15\!\cdots\!02\)\( T^{4} + 7908087460092535224 p^{7} T^{5} + 151767139312 p^{15} T^{6} - 191640 p^{21} T^{7} + p^{28} T^{8} \) |
| 73 | $C_2 \wr S_4$ | \( 1 - 49572 T + 20004112030391 T^{2} - 14597020144465584864 T^{3} + \)\(24\!\cdots\!56\)\( T^{4} - 14597020144465584864 p^{7} T^{5} + 20004112030391 p^{14} T^{6} - 49572 p^{21} T^{7} + p^{28} T^{8} \) |
| 79 | $C_2 \wr S_4$ | \( 1 - 131485 T + 48686541789670 T^{2} + 61012335581189679431 T^{3} + \)\(13\!\cdots\!66\)\( p T^{4} + 61012335581189679431 p^{7} T^{5} + 48686541789670 p^{14} T^{6} - 131485 p^{21} T^{7} + p^{28} T^{8} \) |
| 83 | $C_2 \wr S_4$ | \( 1 - 20201469 T + 255898043168598 T^{2} - \)\(21\!\cdots\!09\)\( T^{3} + \)\(12\!\cdots\!66\)\( T^{4} - \)\(21\!\cdots\!09\)\( p^{7} T^{5} + 255898043168598 p^{14} T^{6} - 20201469 p^{21} T^{7} + p^{28} T^{8} \) |
| 89 | $C_2 \wr S_4$ | \( 1 - 13736753 T + 118323763349406 T^{2} - \)\(82\!\cdots\!11\)\( T^{3} + \)\(58\!\cdots\!66\)\( T^{4} - \)\(82\!\cdots\!11\)\( p^{7} T^{5} + 118323763349406 p^{14} T^{6} - 13736753 p^{21} T^{7} + p^{28} T^{8} \) |
| 97 | $C_2 \wr S_4$ | \( 1 - 572361 T + 270996841471238 T^{2} + 50293597757004117849 T^{3} + \)\(30\!\cdots\!70\)\( T^{4} + 50293597757004117849 p^{7} T^{5} + 270996841471238 p^{14} T^{6} - 572361 p^{21} T^{7} + p^{28} T^{8} \) |
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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
−7.42955114590446062162177246386, −6.79312170291998511255231830416, −6.70336594279772608863003123858, −6.62840219484989600595107174362, −6.43475677155738451574023360573, −5.67775835978411758228597778429, −5.61945263585089244448899422171, −5.55147727070188880225311218821, −5.50769257577380335657217962453, −5.08751890068641731743528658678, −4.75482768780413249711127534880, −4.59691958565150120001985796569, −4.42615783435186266408495397287, −3.62679851013183351859774479848, −3.48920837452075275599233783732, −3.36666530469262391307107303627, −3.14954880764788292313087476056, −2.18064484904407898522429301465, −2.10977205954352902940010493636, −2.06371460119890213217358616732, −2.05257320266784485898864184879, −1.24600256821522559597365486831, −1.11653309146911500731203818482, −0.907905917619363842736531374277, −0.867899445817032938091167622330, 0, 0, 0, 0,
0.867899445817032938091167622330, 0.907905917619363842736531374277, 1.11653309146911500731203818482, 1.24600256821522559597365486831, 2.05257320266784485898864184879, 2.06371460119890213217358616732, 2.10977205954352902940010493636, 2.18064484904407898522429301465, 3.14954880764788292313087476056, 3.36666530469262391307107303627, 3.48920837452075275599233783732, 3.62679851013183351859774479848, 4.42615783435186266408495397287, 4.59691958565150120001985796569, 4.75482768780413249711127534880, 5.08751890068641731743528658678, 5.50769257577380335657217962453, 5.55147727070188880225311218821, 5.61945263585089244448899422171, 5.67775835978411758228597778429, 6.43475677155738451574023360573, 6.62840219484989600595107174362, 6.70336594279772608863003123858, 6.79312170291998511255231830416, 7.42955114590446062162177246386
