| L(s) = 1 | + 3·2-s − 33·4-s − 330·5-s − 686·7-s + 159·8-s − 990·10-s − 2.84e3·11-s + 2.53e3·13-s − 2.05e3·14-s − 1.33e4·16-s + 1.48e3·17-s + 3.28e4·19-s + 1.08e4·20-s − 8.53e3·22-s + 6.57e3·23-s − 5.29e4·25-s + 7.60e3·26-s + 2.26e4·28-s − 2.06e4·29-s − 3.91e5·31-s − 1.08e5·32-s + 4.46e3·34-s + 2.26e5·35-s + 3.67e5·37-s + 9.84e4·38-s − 5.24e4·40-s − 7.34e5·41-s + ⋯ |
| L(s) = 1 | + 0.265·2-s − 0.257·4-s − 1.18·5-s − 0.755·7-s + 0.109·8-s − 0.313·10-s − 0.644·11-s + 0.319·13-s − 0.200·14-s − 0.815·16-s + 0.0734·17-s + 1.09·19-s + 0.304·20-s − 0.170·22-s + 0.112·23-s − 0.677·25-s + 0.0848·26-s + 0.194·28-s − 0.157·29-s − 2.36·31-s − 0.585·32-s + 0.0194·34-s + 0.892·35-s + 1.19·37-s + 0.290·38-s − 0.129·40-s − 1.66·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3969 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3969 ^{s/2} \, \Gamma_{\C}(s+7/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | 3 | | \( 1 \) |
| 7 | $C_1$ | \( ( 1 + p^{3} T )^{2} \) |
| good | 2 | $D_{4}$ | \( 1 - 3 T + 21 p T^{2} - 3 p^{7} T^{3} + p^{14} T^{4} \) |
| 5 | $D_{4}$ | \( 1 + 66 p T + 6474 p^{2} T^{2} + 66 p^{8} T^{3} + p^{14} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + 2844 T + 38086566 T^{2} + 2844 p^{7} T^{3} + p^{14} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - 2534 T - 41123742 T^{2} - 2534 p^{7} T^{3} + p^{14} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 1488 T + 798529822 T^{2} - 1488 p^{7} T^{3} + p^{14} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 32810 T + 1897672038 T^{2} - 32810 p^{7} T^{3} + p^{14} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 6576 T + 6819963598 T^{2} - 6576 p^{7} T^{3} + p^{14} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 20640 T + 15579628518 T^{2} + 20640 p^{7} T^{3} + p^{14} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 391836 T + 92048864606 T^{2} + 391836 p^{7} T^{3} + p^{14} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 367392 T + 63852768182 T^{2} - 367392 p^{7} T^{3} + p^{14} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 734664 T + 402811824126 T^{2} + 734664 p^{7} T^{3} + p^{14} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 480476 T + 594501933318 T^{2} + 480476 p^{7} T^{3} + p^{14} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 1089108 T + 1015337137342 T^{2} - 1089108 p^{7} T^{3} + p^{14} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 2858844 T + 4386858062398 T^{2} + 2858844 p^{7} T^{3} + p^{14} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 160170 T + 4361928868198 T^{2} + 160170 p^{7} T^{3} + p^{14} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 864646 T + 5755969170906 T^{2} + 864646 p^{7} T^{3} + p^{14} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 328648 T + 11587546356582 T^{2} + 328648 p^{7} T^{3} + p^{14} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 7500216 T + 28549732695406 T^{2} - 7500216 p^{7} T^{3} + p^{14} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 4301244 T + 18754109784038 T^{2} - 4301244 p^{7} T^{3} + p^{14} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 6408440 T + 32072611946718 T^{2} + 6408440 p^{7} T^{3} + p^{14} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 11659074 T + 84453675852838 T^{2} + 11659074 p^{7} T^{3} + p^{14} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 9772260 T + 83812995056598 T^{2} + 9772260 p^{7} T^{3} + p^{14} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 10762752 T + 188617737573662 T^{2} - 10762752 p^{7} T^{3} + p^{14} T^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−13.06491888549625883973225242198, −12.81143619718661860143678520322, −12.14428169455207988734786658846, −11.40347463306252400782779289377, −11.20027866166477314488679625802, −10.44200106777677342406415957401, −9.512502218580539658197978710558, −9.357265419574908193838178823983, −8.346556729479870931705821054162, −7.80577094144222461552640318959, −7.25735570773677837075506984262, −6.59180496197993454358641927219, −5.59913548144515842284380765271, −5.04289140062587528421610852258, −4.00240577785679917090978914317, −3.63718965916096938646493471610, −2.74142776833981437015059775183, −1.52034343600579984258394694787, 0, 0,
1.52034343600579984258394694787, 2.74142776833981437015059775183, 3.63718965916096938646493471610, 4.00240577785679917090978914317, 5.04289140062587528421610852258, 5.59913548144515842284380765271, 6.59180496197993454358641927219, 7.25735570773677837075506984262, 7.80577094144222461552640318959, 8.346556729479870931705821054162, 9.357265419574908193838178823983, 9.512502218580539658197978710558, 10.44200106777677342406415957401, 11.20027866166477314488679625802, 11.40347463306252400782779289377, 12.14428169455207988734786658846, 12.81143619718661860143678520322, 13.06491888549625883973225242198
