| 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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 5^{4} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(8-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 5^{4} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+7/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
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
−8.019585204748992677974090968287, −7.39733659982097566542511172980, −7.38752255091673433305240549449, −7.09162077793417704316653823412, −6.97486557741995333062433802734, −6.53821592509485121798549078965, −6.21821202717978136741488090255, −6.17039686086964700385762931339, −6.04730960445422628587498281426, −5.23802203422300502319276072280, −5.16949343089250119228460204085, −5.12157141358602044295894960788, −4.68180069340472076418237796573, −4.18476776333818580076664386723, −4.02481359637926701626747440065, −3.72865845580952587439950444651, −3.59413995225958823704396659107, −3.00288742889126132426066890527, −2.79964375773104627017945618910, −2.76470462007724120029542591898, −2.73467799044445246640032323738, −1.78122461656268523312083201701, −1.38475767879971783076579211620, −1.08134447404803669464735845150, −0.954482005559487061422813699747, 0, 0, 0, 0,
0.954482005559487061422813699747, 1.08134447404803669464735845150, 1.38475767879971783076579211620, 1.78122461656268523312083201701, 2.73467799044445246640032323738, 2.76470462007724120029542591898, 2.79964375773104627017945618910, 3.00288742889126132426066890527, 3.59413995225958823704396659107, 3.72865845580952587439950444651, 4.02481359637926701626747440065, 4.18476776333818580076664386723, 4.68180069340472076418237796573, 5.12157141358602044295894960788, 5.16949343089250119228460204085, 5.23802203422300502319276072280, 6.04730960445422628587498281426, 6.17039686086964700385762931339, 6.21821202717978136741488090255, 6.53821592509485121798549078965, 6.97486557741995333062433802734, 7.09162077793417704316653823412, 7.38752255091673433305240549449, 7.39733659982097566542511172980, 8.019585204748992677974090968287
