| L(s) = 1 | + 32·2-s − 41·3-s + 640·4-s − 363·5-s − 1.31e3·6-s − 774·7-s + 1.02e4·8-s − 4.07e3·9-s − 1.16e4·10-s − 309·11-s − 2.62e4·12-s − 2.08e4·13-s − 2.47e4·14-s + 1.48e4·15-s + 1.43e5·16-s − 4.87e4·17-s − 1.30e5·18-s − 6.90e4·19-s − 2.32e5·20-s + 3.17e4·21-s − 9.88e3·22-s − 5.02e4·23-s − 4.19e5·24-s − 9.21e4·25-s − 6.66e5·26-s + 9.70e4·27-s − 4.95e5·28-s + ⋯ |
| L(s) = 1 | + 2.82·2-s − 0.876·3-s + 5·4-s − 1.29·5-s − 2.47·6-s − 0.852·7-s + 7.07·8-s − 1.86·9-s − 3.67·10-s − 0.0699·11-s − 4.38·12-s − 2.62·13-s − 2.41·14-s + 1.13·15-s + 35/4·16-s − 2.40·17-s − 5.26·18-s − 2.31·19-s − 6.49·20-s + 0.747·21-s − 0.197·22-s − 0.860·23-s − 6.19·24-s − 1.17·25-s − 7.43·26-s + 0.948·27-s − 4.26·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 29986576 ^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 29986576 ^{s/2} \, \Gamma_{\C}(s+7/2)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | $C_1$ | \( ( 1 - p^{3} T )^{4} \) |
| 37 | $C_1$ | \( ( 1 + p^{3} T )^{4} \) |
| good | 3 | $C_2 \wr S_4$ | \( 1 + 41 T + 1918 p T^{2} + 3776 p^{4} T^{3} + 576529 p^{3} T^{4} + 3776 p^{11} T^{5} + 1918 p^{15} T^{6} + 41 p^{21} T^{7} + p^{28} T^{8} \) |
| 5 | $C_2 \wr S_4$ | \( 1 + 363 T + 8957 p^{2} T^{2} + 3150257 p^{2} T^{3} + 191588028 p^{3} T^{4} + 3150257 p^{9} T^{5} + 8957 p^{16} T^{6} + 363 p^{21} T^{7} + p^{28} T^{8} \) |
| 7 | $C_2 \wr S_4$ | \( 1 + 774 T + 2188019 T^{2} + 1491852524 T^{3} + 2319247030460 T^{4} + 1491852524 p^{7} T^{5} + 2188019 p^{14} T^{6} + 774 p^{21} T^{7} + p^{28} T^{8} \) |
| 11 | $C_2 \wr S_4$ | \( 1 + 309 T + 43535464 T^{2} + 54026141408 T^{3} + 1007918833071001 T^{4} + 54026141408 p^{7} T^{5} + 43535464 p^{14} T^{6} + 309 p^{21} T^{7} + p^{28} T^{8} \) |
| 13 | $C_2 \wr S_4$ | \( 1 + 20827 T + 320807181 T^{2} + 3161351859593 T^{3} + 28525048687025980 T^{4} + 3161351859593 p^{7} T^{5} + 320807181 p^{14} T^{6} + 20827 p^{21} T^{7} + p^{28} T^{8} \) |
| 17 | $C_2 \wr S_4$ | \( 1 + 2868 p T + 1953905616 T^{2} + 45703443885900 T^{3} + 1085518000619409886 T^{4} + 45703443885900 p^{7} T^{5} + 1953905616 p^{14} T^{6} + 2868 p^{22} T^{7} + p^{28} T^{8} \) |
| 19 | $C_2 \wr S_4$ | \( 1 + 69068 T + 4225029176 T^{2} + 8930213807348 p T^{3} + 5823792012967010846 T^{4} + 8930213807348 p^{8} T^{5} + 4225029176 p^{14} T^{6} + 69068 p^{21} T^{7} + p^{28} T^{8} \) |
| 23 | $C_2 \wr S_4$ | \( 1 + 50237 T + 3394678469 T^{2} - 3036910301653 p T^{3} + 2973035888453982032 T^{4} - 3036910301653 p^{8} T^{5} + 3394678469 p^{14} T^{6} + 50237 p^{21} T^{7} + p^{28} T^{8} \) |
| 29 | $C_2 \wr S_4$ | \( 1 + 205195 T + 42577610305 T^{2} + 6024089479217461 T^{3} + \)\(73\!\cdots\!28\)\( T^{4} + 6024089479217461 p^{7} T^{5} + 42577610305 p^{14} T^{6} + 205195 p^{21} T^{7} + p^{28} T^{8} \) |
| 31 | $C_2 \wr S_4$ | \( 1 + 172283 T + 120278764723 T^{2} + 14462147611761359 T^{3} + \)\(51\!\cdots\!16\)\( T^{4} + 14462147611761359 p^{7} T^{5} + 120278764723 p^{14} T^{6} + 172283 p^{21} T^{7} + p^{28} T^{8} \) |
| 41 | $C_2 \wr S_4$ | \( 1 + 1018945 T + 1021311629834 T^{2} + 568655619624400024 T^{3} + \)\(31\!\cdots\!01\)\( T^{4} + 568655619624400024 p^{7} T^{5} + 1021311629834 p^{14} T^{6} + 1018945 p^{21} T^{7} + p^{28} T^{8} \) |
| 43 | $C_2 \wr S_4$ | \( 1 - 1263046 T + 1507945247084 T^{2} - 1037351132374985230 T^{3} + \)\(66\!\cdots\!34\)\( T^{4} - 1037351132374985230 p^{7} T^{5} + 1507945247084 p^{14} T^{6} - 1263046 p^{21} T^{7} + p^{28} T^{8} \) |
| 47 | $C_2 \wr S_4$ | \( 1 + 420930 T + 25970100333 p T^{2} + 13737399474630148 p T^{3} + \)\(80\!\cdots\!64\)\( T^{4} + 13737399474630148 p^{8} T^{5} + 25970100333 p^{15} T^{6} + 420930 p^{21} T^{7} + p^{28} T^{8} \) |
| 53 | $C_2 \wr S_4$ | \( 1 - 2051230 T + 3099992511189 T^{2} - 1747572454288272066 T^{3} + \)\(17\!\cdots\!80\)\( T^{4} - 1747572454288272066 p^{7} T^{5} + 3099992511189 p^{14} T^{6} - 2051230 p^{21} T^{7} + p^{28} T^{8} \) |
| 59 | $C_2 \wr S_4$ | \( 1 - 357914 T + 7585596924232 T^{2} - 2187023973339254058 T^{3} + \)\(26\!\cdots\!26\)\( T^{4} - 2187023973339254058 p^{7} T^{5} + 7585596924232 p^{14} T^{6} - 357914 p^{21} T^{7} + p^{28} T^{8} \) |
| 61 | $C_2 \wr S_4$ | \( 1 + 2507513 T + 6763806034565 T^{2} + 179794459198591459 p T^{3} + \)\(22\!\cdots\!48\)\( T^{4} + 179794459198591459 p^{8} T^{5} + 6763806034565 p^{14} T^{6} + 2507513 p^{21} T^{7} + p^{28} T^{8} \) |
| 67 | $C_2 \wr S_4$ | \( 1 - 586879 T + 16166320509849 T^{2} + 19191745994653793 T^{3} + \)\(12\!\cdots\!48\)\( T^{4} + 19191745994653793 p^{7} T^{5} + 16166320509849 p^{14} T^{6} - 586879 p^{21} T^{7} + p^{28} T^{8} \) |
| 71 | $C_2 \wr S_4$ | \( 1 + 130272 T + 14820010273347 T^{2} - 37645872403321354380 T^{3} + \)\(88\!\cdots\!72\)\( T^{4} - 37645872403321354380 p^{7} T^{5} + 14820010273347 p^{14} T^{6} + 130272 p^{21} T^{7} + p^{28} T^{8} \) |
| 73 | $C_2 \wr S_4$ | \( 1 - 3517417 T + 33433888969250 T^{2} - 73386427792716947800 T^{3} + \)\(46\!\cdots\!29\)\( T^{4} - 73386427792716947800 p^{7} T^{5} + 33433888969250 p^{14} T^{6} - 3517417 p^{21} T^{7} + p^{28} T^{8} \) |
| 79 | $C_2 \wr S_4$ | \( 1 - 3790171 T + 55489778352845 T^{2} - \)\(13\!\cdots\!95\)\( T^{3} + \)\(13\!\cdots\!92\)\( T^{4} - \)\(13\!\cdots\!95\)\( p^{7} T^{5} + 55489778352845 p^{14} T^{6} - 3790171 p^{21} T^{7} + p^{28} T^{8} \) |
| 83 | $C_2 \wr S_4$ | \( 1 - 12973460 T + 161107397559259 T^{2} - \)\(11\!\cdots\!40\)\( T^{3} + \)\(73\!\cdots\!16\)\( T^{4} - \)\(11\!\cdots\!40\)\( p^{7} T^{5} + 161107397559259 p^{14} T^{6} - 12973460 p^{21} T^{7} + p^{28} T^{8} \) |
| 89 | $C_2 \wr S_4$ | \( 1 - 18852848 T + 256048437973284 T^{2} - \)\(22\!\cdots\!84\)\( T^{3} + \)\(16\!\cdots\!30\)\( T^{4} - \)\(22\!\cdots\!84\)\( p^{7} T^{5} + 256048437973284 p^{14} T^{6} - 18852848 p^{21} T^{7} + p^{28} T^{8} \) |
| 97 | $C_2 \wr S_4$ | \( 1 - 14580104 T + 241348969368080 T^{2} - \)\(24\!\cdots\!44\)\( T^{3} + \)\(29\!\cdots\!22\)\( T^{4} - \)\(24\!\cdots\!44\)\( p^{7} T^{5} + 241348969368080 p^{14} T^{6} - 14580104 p^{21} T^{7} + p^{28} T^{8} \) |
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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
−10.21718805966398887688658822068, −9.536807797618626744625118999214, −9.438131763266604470600101864000, −8.967081261403608997796964065586, −8.700110102557052583373379688522, −7.948327331037264779109150561429, −7.87819329481920970247719320441, −7.66819007040426517722669485941, −7.27596268098221854896551566867, −6.72621037937633725945670666169, −6.46504463711291201925284819421, −6.25326787060379580892091894981, −6.24030907502542175459363369334, −5.41698340939115048175426358181, −5.31999755850170304929102623674, −5.12237705341799385301014209070, −4.68917329768906413932991237955, −4.20554623498502016499033635429, −3.93188502286257509403606520755, −3.57379220774240847140531508316, −3.52279728217356659471449534741, −2.43487663408383038249081349949, −2.38688653640803562526793719303, −2.32172408316003384122020943012, −1.79135981202518960909468838668, 0, 0, 0, 0,
1.79135981202518960909468838668, 2.32172408316003384122020943012, 2.38688653640803562526793719303, 2.43487663408383038249081349949, 3.52279728217356659471449534741, 3.57379220774240847140531508316, 3.93188502286257509403606520755, 4.20554623498502016499033635429, 4.68917329768906413932991237955, 5.12237705341799385301014209070, 5.31999755850170304929102623674, 5.41698340939115048175426358181, 6.24030907502542175459363369334, 6.25326787060379580892091894981, 6.46504463711291201925284819421, 6.72621037937633725945670666169, 7.27596268098221854896551566867, 7.66819007040426517722669485941, 7.87819329481920970247719320441, 7.948327331037264779109150561429, 8.700110102557052583373379688522, 8.967081261403608997796964065586, 9.438131763266604470600101864000, 9.536807797618626744625118999214, 10.21718805966398887688658822068
