| L(s) = 1 | − 8·2-s + 54·3-s − 92·4-s − 132·5-s − 432·6-s − 1.11e3·7-s + 960·8-s + 2.18e3·9-s + 1.05e3·10-s − 292·11-s − 4.96e3·12-s − 4.39e3·13-s + 8.92e3·14-s − 7.12e3·15-s − 1.52e3·16-s − 1.60e4·17-s − 1.74e4·18-s − 3.48e4·19-s + 1.21e4·20-s − 6.02e4·21-s + 2.33e3·22-s − 8.42e4·23-s + 5.18e4·24-s − 1.43e5·25-s + 3.51e4·26-s + 7.87e4·27-s + 1.02e5·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1.15·3-s − 0.718·4-s − 0.472·5-s − 0.816·6-s − 1.22·7-s + 0.662·8-s + 9-s + 0.333·10-s − 0.0661·11-s − 0.829·12-s − 0.554·13-s + 0.869·14-s − 0.545·15-s − 0.0927·16-s − 0.790·17-s − 0.707·18-s − 1.16·19-s + 0.339·20-s − 1.42·21-s + 0.0467·22-s − 1.44·23-s + 0.765·24-s − 1.83·25-s + 0.392·26-s + 0.769·27-s + 0.883·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1521 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1521 ^{s/2} \, \Gamma_{\C}(s+7/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | $C_1$ | \( ( 1 - p^{3} T )^{2} \) |
| 13 | $C_1$ | \( ( 1 + p^{3} T )^{2} \) |
| good | 2 | $D_{4}$ | \( 1 + p^{3} T + 39 p^{2} T^{2} + p^{10} T^{3} + p^{14} T^{4} \) |
| 5 | $D_{4}$ | \( 1 + 132 T + 32098 p T^{2} + 132 p^{7} T^{3} + p^{14} T^{4} \) |
| 7 | $D_{4}$ | \( 1 + 1116 T + 1885950 T^{2} + 1116 p^{7} T^{3} + p^{14} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + 292 T + 21352058 T^{2} + 292 p^{7} T^{3} + p^{14} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 16004 T + 878784534 T^{2} + 16004 p^{7} T^{3} + p^{14} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 34836 T + 1346863878 T^{2} + 34836 p^{7} T^{3} + p^{14} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 84296 T + 8574812894 T^{2} + 84296 p^{7} T^{3} + p^{14} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 107132 T + 37369002158 T^{2} + 107132 p^{7} T^{3} + p^{14} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 9820 T + 43124615086 T^{2} - 9820 p^{7} T^{3} + p^{14} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 460 p T + 189465655630 T^{2} + 460 p^{8} T^{3} + p^{14} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 542092 T + 453145566978 T^{2} - 542092 p^{7} T^{3} + p^{14} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 260512 T + 377685292150 T^{2} + 260512 p^{7} T^{3} + p^{14} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 1167660 T + 1224388612226 T^{2} - 1167660 p^{7} T^{3} + p^{14} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 22804 T + 1553121851678 T^{2} + 22804 p^{7} T^{3} + p^{14} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 314036 T + 4761555211146 T^{2} + 314036 p^{7} T^{3} + p^{14} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 1240028 T + 4329297452974 T^{2} + 1240028 p^{7} T^{3} + p^{14} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 3256228 T + 6362255889958 T^{2} + 3256228 p^{7} T^{3} + p^{14} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 227516 T + 15764213918882 T^{2} + 227516 p^{7} T^{3} + p^{14} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 1100484 T + 10351835092902 T^{2} - 1100484 p^{7} T^{3} + p^{14} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 12543984 T + 75790244740446 T^{2} + 12543984 p^{7} T^{3} + p^{14} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 2193436 T + 4256986584378 T^{2} + 2193436 p^{7} T^{3} + p^{14} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 4415748 T + 90735340507778 T^{2} + 4415748 p^{7} T^{3} + p^{14} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 4898660 T + 138376132041526 T^{2} - 4898660 p^{7} T^{3} + p^{14} T^{4} \) |
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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
−14.26254596963338886416323141224, −13.89795006482067572175820233651, −13.15543750888371225807347555030, −12.90370054479873281157439398271, −12.13141029273471357403286147929, −11.33207939649125316537896870150, −10.17653568650191007475771220109, −9.999053115370831751637924733489, −9.102712980234007257354708779260, −9.027305990368907120996406950441, −8.017128817958878473642165766235, −7.69054280569805910226920872323, −6.68073841478749291157185565571, −5.84524442341005354145296584522, −4.21410467262449570713187413231, −4.09106944472516766957069114027, −2.84471088229949727828117657614, −1.90228932747277198791997108504, 0, 0,
1.90228932747277198791997108504, 2.84471088229949727828117657614, 4.09106944472516766957069114027, 4.21410467262449570713187413231, 5.84524442341005354145296584522, 6.68073841478749291157185565571, 7.69054280569805910226920872323, 8.017128817958878473642165766235, 9.027305990368907120996406950441, 9.102712980234007257354708779260, 9.999053115370831751637924733489, 10.17653568650191007475771220109, 11.33207939649125316537896870150, 12.13141029273471357403286147929, 12.90370054479873281157439398271, 13.15543750888371225807347555030, 13.89795006482067572175820233651, 14.26254596963338886416323141224
