| L(s) = 1 | − 2·3-s − 3·5-s − 9-s − 5·11-s + 2·13-s + 6·15-s + 10·17-s + 3·19-s − 17·23-s − 25-s + 6·27-s + 2·29-s + 6·31-s + 10·33-s + 14·37-s − 4·39-s + 4·41-s − 17·43-s + 3·45-s − 9·47-s − 20·51-s − 22·53-s + 15·55-s − 6·57-s + 6·59-s + 61-s − 6·65-s + ⋯ |
| L(s) = 1 | − 1.15·3-s − 1.34·5-s − 1/3·9-s − 1.50·11-s + 0.554·13-s + 1.54·15-s + 2.42·17-s + 0.688·19-s − 3.54·23-s − 1/5·25-s + 1.15·27-s + 0.371·29-s + 1.07·31-s + 1.74·33-s + 2.30·37-s − 0.640·39-s + 0.624·41-s − 2.59·43-s + 0.447·45-s − 1.31·47-s − 2.80·51-s − 3.02·53-s + 2.02·55-s − 0.794·57-s + 0.781·59-s + 0.128·61-s − 0.744·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{6} \cdot 7^{6} \cdot 19^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{6} \cdot 7^{6} \cdot 19^{3}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 7 | | \( 1 \) |
| 19 | $C_1$ | \( ( 1 - T )^{3} \) |
| good | 3 | $S_4\times C_2$ | \( 1 + 2 T + 5 T^{2} + 2 p T^{3} + 5 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 5 | $S_4\times C_2$ | \( 1 + 3 T + 2 p T^{2} + 19 T^{3} + 2 p^{2} T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 11 | $S_4\times C_2$ | \( 1 + 5 T + 34 T^{2} + 109 T^{3} + 34 p T^{4} + 5 p^{2} T^{5} + p^{3} T^{6} \) |
| 13 | $S_4\times C_2$ | \( 1 - 2 T + 33 T^{2} - 54 T^{3} + 33 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 17 | $S_4\times C_2$ | \( 1 - 10 T + 67 T^{2} - 304 T^{3} + 67 p T^{4} - 10 p^{2} T^{5} + p^{3} T^{6} \) |
| 23 | $S_4\times C_2$ | \( 1 + 17 T + 160 T^{2} + 931 T^{3} + 160 p T^{4} + 17 p^{2} T^{5} + p^{3} T^{6} \) |
| 29 | $S_4\times C_2$ | \( 1 - 2 T + p T^{2} - 134 T^{3} + p^{2} T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 31 | $S_4\times C_2$ | \( 1 - 6 T + 57 T^{2} - 358 T^{3} + 57 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 37 | $S_4\times C_2$ | \( 1 - 14 T + 169 T^{2} - 30 p T^{3} + 169 p T^{4} - 14 p^{2} T^{5} + p^{3} T^{6} \) |
| 41 | $S_4\times C_2$ | \( 1 - 4 T + 51 T^{2} - 472 T^{3} + 51 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 43 | $S_4\times C_2$ | \( 1 + 17 T + 220 T^{2} + 1611 T^{3} + 220 p T^{4} + 17 p^{2} T^{5} + p^{3} T^{6} \) |
| 47 | $S_4\times C_2$ | \( 1 + 9 T + 124 T^{2} + 735 T^{3} + 124 p T^{4} + 9 p^{2} T^{5} + p^{3} T^{6} \) |
| 53 | $S_4\times C_2$ | \( 1 + 22 T + 225 T^{2} + 1706 T^{3} + 225 p T^{4} + 22 p^{2} T^{5} + p^{3} T^{6} \) |
| 59 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{3} \) |
| 61 | $S_4\times C_2$ | \( 1 - T + 126 T^{2} - 261 T^{3} + 126 p T^{4} - p^{2} T^{5} + p^{3} T^{6} \) |
| 67 | $S_4\times C_2$ | \( 1 - 2 T + 133 T^{2} - 100 T^{3} + 133 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 71 | $S_4\times C_2$ | \( 1 + 18 T + 277 T^{2} + 2514 T^{3} + 277 p T^{4} + 18 p^{2} T^{5} + p^{3} T^{6} \) |
| 73 | $S_4\times C_2$ | \( 1 + 21 T + 334 T^{2} + 3217 T^{3} + 334 p T^{4} + 21 p^{2} T^{5} + p^{3} T^{6} \) |
| 79 | $S_4\times C_2$ | \( 1 - 8 T + 253 T^{2} - 1266 T^{3} + 253 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 83 | $S_4\times C_2$ | \( 1 - 27 T + 426 T^{2} - 4439 T^{3} + 426 p T^{4} - 27 p^{2} T^{5} + p^{3} T^{6} \) |
| 89 | $S_4\times C_2$ | \( 1 + 30 T + 501 T^{2} + 5502 T^{3} + 501 p T^{4} + 30 p^{2} T^{5} + p^{3} T^{6} \) |
| 97 | $S_4\times C_2$ | \( 1 - 14 T + 119 T^{2} - 1252 T^{3} + 119 p T^{4} - 14 p^{2} T^{5} + p^{3} T^{6} \) |
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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
−7.917549441894300742215295704536, −7.77771017374071011845873818023, −7.65810837774808098603254207517, −7.34406016692015463907470718202, −6.78725180752673093247316145623, −6.62396745751195080914907822486, −6.33748917171275030514348238141, −6.02714920013909723476670684605, −5.82955778538108564668481829748, −5.79473509243198593522896058360, −5.48053005281760321786826149591, −5.17726112097458194661994816131, −4.91293094674118048737648928705, −4.55255476816345666543820639882, −4.34564161666153776001083134109, −4.16932521480383459554958117862, −3.59756913058653986068066611719, −3.43146768360449209622122638107, −3.34830595301974350672237773885, −2.80244496267558262259992510542, −2.67345445942629267619065552653, −2.25012553492098650550712536626, −1.64139562653932564803890141457, −1.30178853215403316268020547367, −1.02303263923259426795241948509, 0, 0, 0,
1.02303263923259426795241948509, 1.30178853215403316268020547367, 1.64139562653932564803890141457, 2.25012553492098650550712536626, 2.67345445942629267619065552653, 2.80244496267558262259992510542, 3.34830595301974350672237773885, 3.43146768360449209622122638107, 3.59756913058653986068066611719, 4.16932521480383459554958117862, 4.34564161666153776001083134109, 4.55255476816345666543820639882, 4.91293094674118048737648928705, 5.17726112097458194661994816131, 5.48053005281760321786826149591, 5.79473509243198593522896058360, 5.82955778538108564668481829748, 6.02714920013909723476670684605, 6.33748917171275030514348238141, 6.62396745751195080914907822486, 6.78725180752673093247316145623, 7.34406016692015463907470718202, 7.65810837774808098603254207517, 7.77771017374071011845873818023, 7.917549441894300742215295704536
