| L(s) = 1 | + 3-s − 500·5-s + 1.37e3·7-s − 2.58e3·9-s + 465·11-s + 5.53e3·13-s − 500·15-s − 1.83e4·17-s − 2.79e4·19-s + 1.37e3·21-s − 5.72e4·23-s + 1.56e5·25-s − 6.31e4·27-s + 3.04e4·29-s − 1.36e5·31-s + 465·33-s − 6.86e5·35-s − 8.77e4·37-s + 5.53e3·39-s − 5.92e5·41-s + 1.88e5·43-s + 1.29e6·45-s + 4.85e5·47-s + 1.17e6·49-s − 1.83e4·51-s + 6.33e5·53-s − 2.32e5·55-s + ⋯ |
| L(s) = 1 | + 0.0213·3-s − 1.78·5-s + 1.51·7-s − 1.18·9-s + 0.105·11-s + 0.698·13-s − 0.0382·15-s − 0.904·17-s − 0.934·19-s + 0.0323·21-s − 0.981·23-s + 2·25-s − 0.617·27-s + 0.232·29-s − 0.823·31-s + 0.00225·33-s − 2.70·35-s − 0.284·37-s + 0.0149·39-s − 1.34·41-s + 0.361·43-s + 2.11·45-s + 0.682·47-s + 10/7·49-s − 0.0193·51-s + 0.584·53-s − 0.188·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 5^{4} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(8-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 5^{4} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+7/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
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 | | \( 1 \) |
| 5 | $C_1$ | \( ( 1 + p^{3} T )^{4} \) |
| 7 | $C_1$ | \( ( 1 - p^{3} T )^{4} \) |
| good | 3 | $C_2 \wr S_4$ | \( 1 - T + 287 p^{2} T^{2} + 716 p^{4} T^{3} + 232472 p^{3} T^{4} + 716 p^{11} T^{5} + 287 p^{16} T^{6} - p^{21} T^{7} + p^{28} T^{8} \) |
| 11 | $C_2 \wr S_4$ | \( 1 - 465 T + 73977203 T^{2} - 26499720020 T^{3} + 2123902952289324 T^{4} - 26499720020 p^{7} T^{5} + 73977203 p^{14} T^{6} - 465 p^{21} T^{7} + p^{28} T^{8} \) |
| 13 | $C_2 \wr S_4$ | \( 1 - 5537 T + 234952765 T^{2} - 973254991602 T^{3} + 1664419765898034 p T^{4} - 973254991602 p^{7} T^{5} + 234952765 p^{14} T^{6} - 5537 p^{21} T^{7} + p^{28} T^{8} \) |
| 17 | $C_2 \wr S_4$ | \( 1 + 18325 T + 1036193681 T^{2} + 24810867687838 T^{3} + 512631106693527190 T^{4} + 24810867687838 p^{7} T^{5} + 1036193681 p^{14} T^{6} + 18325 p^{21} T^{7} + p^{28} T^{8} \) |
| 19 | $C_2 \wr S_4$ | \( 1 + 1470 p T + 2698932472 T^{2} + 3375718121734 p T^{3} + 3436931013554175006 T^{4} + 3375718121734 p^{8} T^{5} + 2698932472 p^{14} T^{6} + 1470 p^{22} T^{7} + p^{28} T^{8} \) |
| 23 | $C_2 \wr S_4$ | \( 1 + 57282 T + 14108203976 T^{2} + 575555998365506 T^{3} + 72802700173827715086 T^{4} + 575555998365506 p^{7} T^{5} + 14108203976 p^{14} T^{6} + 57282 p^{21} T^{7} + p^{28} T^{8} \) |
| 29 | $C_2 \wr S_4$ | \( 1 - 30471 T + 34087437629 T^{2} - 2806477165335966 T^{3} + \)\(60\!\cdots\!66\)\( T^{4} - 2806477165335966 p^{7} T^{5} + 34087437629 p^{14} T^{6} - 30471 p^{21} T^{7} + p^{28} T^{8} \) |
| 31 | $C_2 \wr S_4$ | \( 1 + 136600 T + 88869658108 T^{2} + 11450410777968568 T^{3} + \)\(33\!\cdots\!26\)\( T^{4} + 11450410777968568 p^{7} T^{5} + 88869658108 p^{14} T^{6} + 136600 p^{21} T^{7} + p^{28} T^{8} \) |
| 37 | $C_2 \wr S_4$ | \( 1 + 87704 T + 141264114220 T^{2} - 5215893402731128 T^{3} + \)\(14\!\cdots\!58\)\( T^{4} - 5215893402731128 p^{7} T^{5} + 141264114220 p^{14} T^{6} + 87704 p^{21} T^{7} + p^{28} T^{8} \) |
| 41 | $C_2 \wr S_4$ | \( 1 + 592022 T + 585021288980 T^{2} + 163170796094716018 T^{3} + \)\(12\!\cdots\!58\)\( T^{4} + 163170796094716018 p^{7} T^{5} + 585021288980 p^{14} T^{6} + 592022 p^{21} T^{7} + p^{28} T^{8} \) |
| 43 | $C_2 \wr S_4$ | \( 1 - 188510 T + 350783002720 T^{2} + 216069010760756682 T^{3} - \)\(29\!\cdots\!58\)\( T^{4} + 216069010760756682 p^{7} T^{5} + 350783002720 p^{14} T^{6} - 188510 p^{21} T^{7} + p^{28} T^{8} \) |
| 47 | $C_2 \wr S_4$ | \( 1 - 485793 T + 1003699259135 T^{2} - 273376866489758352 T^{3} + \)\(45\!\cdots\!80\)\( T^{4} - 273376866489758352 p^{7} T^{5} + 1003699259135 p^{14} T^{6} - 485793 p^{21} T^{7} + p^{28} T^{8} \) |
| 53 | $C_2 \wr S_4$ | \( 1 - 633074 T + 3930175503212 T^{2} - 1747926630355958494 T^{3} + \)\(64\!\cdots\!30\)\( T^{4} - 1747926630355958494 p^{7} T^{5} + 3930175503212 p^{14} T^{6} - 633074 p^{21} T^{7} + p^{28} T^{8} \) |
| 59 | $C_2 \wr S_4$ | \( 1 - 1937648 T + 9173351204684 T^{2} - 12936943934523224432 T^{3} + \)\(33\!\cdots\!30\)\( T^{4} - 12936943934523224432 p^{7} T^{5} + 9173351204684 p^{14} T^{6} - 1937648 p^{21} T^{7} + p^{28} T^{8} \) |
| 61 | $C_2 \wr S_4$ | \( 1 + 2013762 T + 13785159581164 T^{2} + 19215948602053024734 T^{3} + \)\(66\!\cdots\!90\)\( T^{4} + 19215948602053024734 p^{7} T^{5} + 13785159581164 p^{14} T^{6} + 2013762 p^{21} T^{7} + p^{28} T^{8} \) |
| 67 | $C_2 \wr S_4$ | \( 1 - 963240 T + 20196657485164 T^{2} - 14210228329725610792 T^{3} + \)\(17\!\cdots\!06\)\( T^{4} - 14210228329725610792 p^{7} T^{5} + 20196657485164 p^{14} T^{6} - 963240 p^{21} T^{7} + p^{28} T^{8} \) |
| 71 | $C_2 \wr S_4$ | \( 1 - 5159784 T + 30402046058012 T^{2} - \)\(12\!\cdots\!92\)\( T^{3} + \)\(40\!\cdots\!22\)\( T^{4} - \)\(12\!\cdots\!92\)\( p^{7} T^{5} + 30402046058012 p^{14} T^{6} - 5159784 p^{21} T^{7} + p^{28} T^{8} \) |
| 73 | $C_2 \wr S_4$ | \( 1 - 2062816 T + 17101834108492 T^{2} - 32128189785598381536 T^{3} + \)\(16\!\cdots\!30\)\( T^{4} - 32128189785598381536 p^{7} T^{5} + 17101834108492 p^{14} T^{6} - 2062816 p^{21} T^{7} + p^{28} T^{8} \) |
| 79 | $C_2 \wr S_4$ | \( 1 - 5024559 T + 70934378110879 T^{2} - \)\(25\!\cdots\!84\)\( T^{3} + \)\(20\!\cdots\!36\)\( T^{4} - \)\(25\!\cdots\!84\)\( p^{7} T^{5} + 70934378110879 p^{14} T^{6} - 5024559 p^{21} T^{7} + p^{28} T^{8} \) |
| 83 | $C_2 \wr S_4$ | \( 1 - 3418580 T + 64277753407484 T^{2} - \)\(33\!\cdots\!96\)\( T^{3} + \)\(20\!\cdots\!30\)\( T^{4} - \)\(33\!\cdots\!96\)\( p^{7} T^{5} + 64277753407484 p^{14} T^{6} - 3418580 p^{21} T^{7} + p^{28} T^{8} \) |
| 89 | $C_2 \wr S_4$ | \( 1 + 3740246 T + 1794452330596 p T^{2} + \)\(43\!\cdots\!42\)\( T^{3} + \)\(10\!\cdots\!10\)\( T^{4} + \)\(43\!\cdots\!42\)\( p^{7} T^{5} + 1794452330596 p^{15} T^{6} + 3740246 p^{21} T^{7} + p^{28} T^{8} \) |
| 97 | $C_2 \wr S_4$ | \( 1 + 14663513 T + 86644715122369 T^{2} - \)\(83\!\cdots\!10\)\( T^{3} - \)\(12\!\cdots\!18\)\( T^{4} - \)\(83\!\cdots\!10\)\( p^{7} T^{5} + 86644715122369 p^{14} T^{6} + 14663513 p^{21} T^{7} + p^{28} T^{8} \) |
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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
−7.05872073928719378889418352669, −6.78026697872693748360636544572, −6.56612470741244760575249670137, −6.52467171336249579797950386475, −6.29231964493469308544859399718, −5.63990693494688293573479609124, −5.45158640731404013687854106784, −5.42323166986776420248996572460, −5.39431550205902946053350741544, −4.74725435739852034149392827141, −4.53007210456345288943543840253, −4.39295355252359598198534430375, −4.19635599642623849899696848827, −3.83505165818429181152276519275, −3.58072008446414769121411657565, −3.47882167794824655397037930848, −3.33855419685317176164824583416, −2.54748954805667385637278421751, −2.41880583983011758443304249120, −2.35796684827081139444865265994, −2.11588201259413789879724753243, −1.51234214032774799304141230430, −1.31048094728749114474334269754, −1.03613677375602910848541996030, −0.871086179764514992230067969130, 0, 0, 0, 0,
0.871086179764514992230067969130, 1.03613677375602910848541996030, 1.31048094728749114474334269754, 1.51234214032774799304141230430, 2.11588201259413789879724753243, 2.35796684827081139444865265994, 2.41880583983011758443304249120, 2.54748954805667385637278421751, 3.33855419685317176164824583416, 3.47882167794824655397037930848, 3.58072008446414769121411657565, 3.83505165818429181152276519275, 4.19635599642623849899696848827, 4.39295355252359598198534430375, 4.53007210456345288943543840253, 4.74725435739852034149392827141, 5.39431550205902946053350741544, 5.42323166986776420248996572460, 5.45158640731404013687854106784, 5.63990693494688293573479609124, 6.29231964493469308544859399718, 6.52467171336249579797950386475, 6.56612470741244760575249670137, 6.78026697872693748360636544572, 7.05872073928719378889418352669
