| L(s) = 1 | − 3·2-s − 3-s + 3·4-s + 3·6-s + 35·7-s − 8-s − 16·9-s + 16·11-s − 3·12-s − 65·13-s − 105·14-s − 33·16-s − 29·17-s + 48·18-s − 57·19-s − 35·21-s − 48·22-s + 101·23-s + 24-s + 195·26-s + 151·27-s + 105·28-s + 377·29-s − 140·31-s + 27·32-s − 16·33-s + 87·34-s + ⋯ |
| L(s) = 1 | − 1.06·2-s − 0.192·3-s + 3/8·4-s + 0.204·6-s + 1.88·7-s − 0.0441·8-s − 0.592·9-s + 0.438·11-s − 0.0721·12-s − 1.38·13-s − 2.00·14-s − 0.515·16-s − 0.413·17-s + 0.628·18-s − 0.688·19-s − 0.363·21-s − 0.465·22-s + 0.915·23-s + 0.00850·24-s + 1.47·26-s + 1.07·27-s + 0.708·28-s + 2.41·29-s − 0.811·31-s + 0.149·32-s − 0.0844·33-s + 0.438·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{6} \cdot 19^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{6} \cdot 19^{3}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.989146483\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.989146483\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 5 | | \( 1 \) |
| 19 | $C_1$ | \( ( 1 + p T )^{3} \) |
| good | 2 | $S_4\times C_2$ | \( 1 + 3 T + 3 p T^{2} + 5 p T^{3} + 3 p^{4} T^{4} + 3 p^{6} T^{5} + p^{9} T^{6} \) |
| 3 | $S_4\times C_2$ | \( 1 + T + 17 T^{2} - 118 T^{3} + 17 p^{3} T^{4} + p^{6} T^{5} + p^{9} T^{6} \) |
| 7 | $S_4\times C_2$ | \( 1 - 5 p T + 24 p^{2} T^{2} - 21691 T^{3} + 24 p^{5} T^{4} - 5 p^{7} T^{5} + p^{9} T^{6} \) |
| 11 | $S_4\times C_2$ | \( 1 - 16 T + 3942 T^{2} - 41410 T^{3} + 3942 p^{3} T^{4} - 16 p^{6} T^{5} + p^{9} T^{6} \) |
| 13 | $S_4\times C_2$ | \( 1 + 5 p T + 7335 T^{2} + 280762 T^{3} + 7335 p^{3} T^{4} + 5 p^{7} T^{5} + p^{9} T^{6} \) |
| 17 | $S_4\times C_2$ | \( 1 + 29 T + 5514 T^{2} + 503573 T^{3} + 5514 p^{3} T^{4} + 29 p^{6} T^{5} + p^{9} T^{6} \) |
| 23 | $S_4\times C_2$ | \( 1 - 101 T + 31877 T^{2} - 2079558 T^{3} + 31877 p^{3} T^{4} - 101 p^{6} T^{5} + p^{9} T^{6} \) |
| 29 | $S_4\times C_2$ | \( 1 - 13 p T + 81935 T^{2} - 13844910 T^{3} + 81935 p^{3} T^{4} - 13 p^{7} T^{5} + p^{9} T^{6} \) |
| 31 | $S_4\times C_2$ | \( 1 + 140 T + 51757 T^{2} + 5897128 T^{3} + 51757 p^{3} T^{4} + 140 p^{6} T^{5} + p^{9} T^{6} \) |
| 37 | $S_4\times C_2$ | \( 1 - 290 T + 105187 T^{2} - 19377292 T^{3} + 105187 p^{3} T^{4} - 290 p^{6} T^{5} + p^{9} T^{6} \) |
| 41 | $S_4\times C_2$ | \( 1 - 956 T + 508879 T^{2} - 163355096 T^{3} + 508879 p^{3} T^{4} - 956 p^{6} T^{5} + p^{9} T^{6} \) |
| 43 | $S_4\times C_2$ | \( 1 - 570 T + 145938 T^{2} - 24674476 T^{3} + 145938 p^{3} T^{4} - 570 p^{6} T^{5} + p^{9} T^{6} \) |
| 47 | $S_4\times C_2$ | \( 1 + 66 T + 280158 T^{2} + 10764012 T^{3} + 280158 p^{3} T^{4} + 66 p^{6} T^{5} + p^{9} T^{6} \) |
| 53 | $S_4\times C_2$ | \( 1 + 817 T + 657711 T^{2} + 260089834 T^{3} + 657711 p^{3} T^{4} + 817 p^{6} T^{5} + p^{9} T^{6} \) |
| 59 | $S_4\times C_2$ | \( 1 - 265 T + 458145 T^{2} - 77293258 T^{3} + 458145 p^{3} T^{4} - 265 p^{6} T^{5} + p^{9} T^{6} \) |
| 61 | $S_4\times C_2$ | \( 1 - 988 T + 726644 T^{2} - 371638582 T^{3} + 726644 p^{3} T^{4} - 988 p^{6} T^{5} + p^{9} T^{6} \) |
| 67 | $S_4\times C_2$ | \( 1 - 207 T + 842361 T^{2} - 117000634 T^{3} + 842361 p^{3} T^{4} - 207 p^{6} T^{5} + p^{9} T^{6} \) |
| 71 | $S_4\times C_2$ | \( 1 - 846 T + 1246593 T^{2} - 603857484 T^{3} + 1246593 p^{3} T^{4} - 846 p^{6} T^{5} + p^{9} T^{6} \) |
| 73 | $S_4\times C_2$ | \( 1 + 627 T + 811566 T^{2} + 342245479 T^{3} + 811566 p^{3} T^{4} + 627 p^{6} T^{5} + p^{9} T^{6} \) |
| 79 | $S_4\times C_2$ | \( 1 - 382 T + 809229 T^{2} - 432705284 T^{3} + 809229 p^{3} T^{4} - 382 p^{6} T^{5} + p^{9} T^{6} \) |
| 83 | $S_4\times C_2$ | \( 1 - 766 T + 1679713 T^{2} - 797249332 T^{3} + 1679713 p^{3} T^{4} - 766 p^{6} T^{5} + p^{9} T^{6} \) |
| 89 | $S_4\times C_2$ | \( 1 + 172 T + 1270123 T^{2} + 165585880 T^{3} + 1270123 p^{3} T^{4} + 172 p^{6} T^{5} + p^{9} T^{6} \) |
| 97 | $S_4\times C_2$ | \( 1 - 2450 T + 4122563 T^{2} - 4668536612 T^{3} + 4122563 p^{3} T^{4} - 2450 p^{6} T^{5} + p^{9} T^{6} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{6} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.436482581375267902912533771777, −9.043350233772598800900331803727, −8.762422921597023549809201507134, −8.659694557468588902537564026628, −8.074038146281281106556226935227, −8.017505772753163634616575100963, −7.57002309337987325001596970058, −7.54716449145074726078191911488, −6.97102109035760990027347337903, −6.61675634312065184113569977842, −6.38062699982005776042183722939, −5.93973126883544044364403939594, −5.63100643969690231456477576396, −4.91283871887317861925698601151, −4.89325207441706797424285493401, −4.66034568682703804988250649926, −4.25640372100472110897810053270, −3.86676459926048703157287418771, −3.03238743106983099750173163075, −2.68830346625041858979088412528, −2.17075433587517844871797033346, −2.15506696931849794623376472593, −1.08247765685109220844021774056, −0.981616591833813509542561010687, −0.42241668340230481051776815580,
0.42241668340230481051776815580, 0.981616591833813509542561010687, 1.08247765685109220844021774056, 2.15506696931849794623376472593, 2.17075433587517844871797033346, 2.68830346625041858979088412528, 3.03238743106983099750173163075, 3.86676459926048703157287418771, 4.25640372100472110897810053270, 4.66034568682703804988250649926, 4.89325207441706797424285493401, 4.91283871887317861925698601151, 5.63100643969690231456477576396, 5.93973126883544044364403939594, 6.38062699982005776042183722939, 6.61675634312065184113569977842, 6.97102109035760990027347337903, 7.54716449145074726078191911488, 7.57002309337987325001596970058, 8.017505772753163634616575100963, 8.074038146281281106556226935227, 8.659694557468588902537564026628, 8.762422921597023549809201507134, 9.043350233772598800900331803727, 9.436482581375267902912533771777
