| L(s) = 1 | + 3·2-s + 2·4-s − 15·5-s + 10·7-s − 3·8-s − 45·10-s − 24·11-s − 2·13-s + 30·14-s − 3·16-s − 2·19-s − 30·20-s − 72·22-s − 15·23-s + 125·25-s − 6·26-s + 20·28-s + 21·29-s − 20·31-s − 12·32-s − 150·35-s − 56·37-s − 6·38-s + 45·40-s − 24·41-s − 62·43-s − 48·44-s + ⋯ |
| L(s) = 1 | + 3/2·2-s + 1/2·4-s − 3·5-s + 10/7·7-s − 3/8·8-s − 9/2·10-s − 2.18·11-s − 0.153·13-s + 15/7·14-s − 0.187·16-s − 0.105·19-s − 3/2·20-s − 3.27·22-s − 0.652·23-s + 5·25-s − 0.230·26-s + 5/7·28-s + 0.724·29-s − 0.645·31-s − 3/8·32-s − 4.28·35-s − 1.51·37-s − 0.157·38-s + 9/8·40-s − 0.585·41-s − 1.44·43-s − 1.09·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 59049 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 59049 ^{s/2} \, \Gamma_{\C}(s+1)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.112977259\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.112977259\) |
| \(L(2)\) |
|
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 | | \( 1 \) |
| good | 2 | $C_2^2$ | \( 1 - 3 T + 7 T^{2} - 3 p^{2} T^{3} + p^{4} T^{4} \) |
| 5 | $C_1$$\times$$C_2$ | \( ( 1 + p T )^{2}( 1 + p T + p^{2} T^{2} ) \) |
| 7 | $C_2^2$ | \( 1 - 10 T + 51 T^{2} - 10 p^{2} T^{3} + p^{4} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 24 T + 313 T^{2} + 24 p^{2} T^{3} + p^{4} T^{4} \) |
| 13 | $C_2^2$ | \( 1 + 2 T - 165 T^{2} + 2 p^{2} T^{3} + p^{4} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 470 T^{2} + p^{4} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + T + p^{2} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 + 15 T + 604 T^{2} + 15 p^{2} T^{3} + p^{4} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 21 T + 988 T^{2} - 21 p^{2} T^{3} + p^{4} T^{4} \) |
| 31 | $C_2^2$ | \( 1 + 20 T - 561 T^{2} + 20 p^{2} T^{3} + p^{4} T^{4} \) |
| 37 | $C_2$ | \( ( 1 + 28 T + p^{2} T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 + 24 T + 1873 T^{2} + 24 p^{2} T^{3} + p^{4} T^{4} \) |
| 43 | $C_2^2$ | \( 1 + 62 T + 1995 T^{2} + 62 p^{2} T^{3} + p^{4} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 93 T + 5092 T^{2} - 93 p^{2} T^{3} + p^{4} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 1055 T^{2} + p^{4} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 66 T + 4933 T^{2} - 66 p^{2} T^{3} + p^{4} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 34 T - 2565 T^{2} - 34 p^{2} T^{3} + p^{4} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 5 T - 4464 T^{2} + 5 p^{2} T^{3} + p^{4} T^{4} \) |
| 71 | $C_2^2$ | \( 1 - 10055 T^{2} + p^{4} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 137 T + p^{2} T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 + 80 T + 159 T^{2} + 80 p^{2} T^{3} + p^{4} T^{4} \) |
| 83 | $C_2^2$ | \( 1 - 120 T + 11689 T^{2} - 120 p^{2} T^{3} + p^{4} T^{4} \) |
| 89 | $C_2^2$ | \( 1 + 8458 T^{2} + p^{4} T^{4} \) |
| 97 | $C_2^2$ | \( 1 + 17 T - 9120 T^{2} + 17 p^{2} T^{3} + p^{4} T^{4} \) |
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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
−12.25010844620087369405750146056, −11.67772434463158152363836680091, −11.46352640274224493205074757454, −10.84329611286192506556137410835, −10.61754055810147484548078652672, −10.01711684930875242974015312672, −8.801826681632423345683555830146, −8.354974032637061007982361996039, −8.189266986890722782145298478797, −7.50614466264138464406037884242, −7.49790695710384172454927139731, −6.69037024060955413525342318979, −5.39428666028600817467827576703, −5.17662276271383374752745810065, −4.77993895542070262915307961210, −4.20923863597470108327135352683, −3.73407826202446150033603096743, −3.27539057862633224302909101759, −2.24667305657941093531123060353, −0.43607992286639274612823851598,
0.43607992286639274612823851598, 2.24667305657941093531123060353, 3.27539057862633224302909101759, 3.73407826202446150033603096743, 4.20923863597470108327135352683, 4.77993895542070262915307961210, 5.17662276271383374752745810065, 5.39428666028600817467827576703, 6.69037024060955413525342318979, 7.49790695710384172454927139731, 7.50614466264138464406037884242, 8.189266986890722782145298478797, 8.354974032637061007982361996039, 8.801826681632423345683555830146, 10.01711684930875242974015312672, 10.61754055810147484548078652672, 10.84329611286192506556137410835, 11.46352640274224493205074757454, 11.67772434463158152363836680091, 12.25010844620087369405750146056
