| L(s) = 1 | − 9·3-s − 15·5-s − 21·7-s + 54·9-s − 6·11-s + 36·13-s + 135·15-s + 90·17-s + 78·19-s + 189·21-s + 150·25-s − 270·27-s + 162·29-s + 114·31-s + 54·33-s + 315·35-s + 246·37-s − 324·39-s − 42·41-s − 348·43-s − 810·45-s − 480·47-s + 294·49-s − 810·51-s − 252·53-s + 90·55-s − 702·57-s + ⋯ |
| L(s) = 1 | − 1.73·3-s − 1.34·5-s − 1.13·7-s + 2·9-s − 0.164·11-s + 0.768·13-s + 2.32·15-s + 1.28·17-s + 0.941·19-s + 1.96·21-s + 6/5·25-s − 1.92·27-s + 1.03·29-s + 0.660·31-s + 0.284·33-s + 1.52·35-s + 1.09·37-s − 1.33·39-s − 0.159·41-s − 1.23·43-s − 2.68·45-s − 1.48·47-s + 6/7·49-s − 2.22·51-s − 0.653·53-s + 0.220·55-s − 1.63·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{9} \cdot 3^{3} \cdot 5^{3} \cdot 7^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{9} \cdot 3^{3} \cdot 5^{3} \cdot 7^{3}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | | \( 1 \) |
| 3 | $C_1$ | \( ( 1 + p T )^{3} \) |
| 5 | $C_1$ | \( ( 1 + p T )^{3} \) |
| 7 | $C_1$ | \( ( 1 + p T )^{3} \) |
| good | 11 | $S_4\times C_2$ | \( 1 + 6 T + 849 T^{2} + 58724 T^{3} + 849 p^{3} T^{4} + 6 p^{6} T^{5} + p^{9} T^{6} \) |
| 13 | $S_4\times C_2$ | \( 1 - 36 T + 3867 T^{2} - 171096 T^{3} + 3867 p^{3} T^{4} - 36 p^{6} T^{5} + p^{9} T^{6} \) |
| 17 | $S_4\times C_2$ | \( 1 - 90 T + 7215 T^{2} - 209228 T^{3} + 7215 p^{3} T^{4} - 90 p^{6} T^{5} + p^{9} T^{6} \) |
| 19 | $S_4\times C_2$ | \( 1 - 78 T - 615 T^{2} + 858188 T^{3} - 615 p^{3} T^{4} - 78 p^{6} T^{5} + p^{9} T^{6} \) |
| 23 | $S_4\times C_2$ | \( 1 + 29013 T^{2} - 174080 T^{3} + 29013 p^{3} T^{4} + p^{9} T^{6} \) |
| 29 | $S_4\times C_2$ | \( 1 - 162 T + 69291 T^{2} - 6985356 T^{3} + 69291 p^{3} T^{4} - 162 p^{6} T^{5} + p^{9} T^{6} \) |
| 31 | $S_4\times C_2$ | \( 1 - 114 T + 88005 T^{2} - 6676828 T^{3} + 88005 p^{3} T^{4} - 114 p^{6} T^{5} + p^{9} T^{6} \) |
| 37 | $S_4\times C_2$ | \( 1 - 246 T + 104739 T^{2} - 15246340 T^{3} + 104739 p^{3} T^{4} - 246 p^{6} T^{5} + p^{9} T^{6} \) |
| 41 | $S_4\times C_2$ | \( 1 + 42 T + 115911 T^{2} + 5099596 T^{3} + 115911 p^{3} T^{4} + 42 p^{6} T^{5} + p^{9} T^{6} \) |
| 43 | $S_4\times C_2$ | \( 1 + 348 T - 31431 T^{2} - 43144536 T^{3} - 31431 p^{3} T^{4} + 348 p^{6} T^{5} + p^{9} T^{6} \) |
| 47 | $S_4\times C_2$ | \( 1 + 480 T + 358269 T^{2} + 97277376 T^{3} + 358269 p^{3} T^{4} + 480 p^{6} T^{5} + p^{9} T^{6} \) |
| 53 | $S_4\times C_2$ | \( 1 + 252 T + 270339 T^{2} + 25739624 T^{3} + 270339 p^{3} T^{4} + 252 p^{6} T^{5} + p^{9} T^{6} \) |
| 59 | $S_4\times C_2$ | \( 1 + 408 T + 227193 T^{2} - 3891952 T^{3} + 227193 p^{3} T^{4} + 408 p^{6} T^{5} + p^{9} T^{6} \) |
| 61 | $S_4\times C_2$ | \( 1 + 534 T + 573099 T^{2} + 246084228 T^{3} + 573099 p^{3} T^{4} + 534 p^{6} T^{5} + p^{9} T^{6} \) |
| 67 | $S_4\times C_2$ | \( 1 + 372 T + 387249 T^{2} + 89969272 T^{3} + 387249 p^{3} T^{4} + 372 p^{6} T^{5} + p^{9} T^{6} \) |
| 71 | $S_4\times C_2$ | \( 1 + 366 T + 978093 T^{2} + 238426052 T^{3} + 978093 p^{3} T^{4} + 366 p^{6} T^{5} + p^{9} T^{6} \) |
| 73 | $S_4\times C_2$ | \( 1 + 60 T + 284871 T^{2} + 81788808 T^{3} + 284871 p^{3} T^{4} + 60 p^{6} T^{5} + p^{9} T^{6} \) |
| 79 | $S_4\times C_2$ | \( 1 + 624 T + 1002381 T^{2} + 4699360 p T^{3} + 1002381 p^{3} T^{4} + 624 p^{6} T^{5} + p^{9} T^{6} \) |
| 83 | $S_4\times C_2$ | \( 1 + 3228 T + 5037969 T^{2} + 4755643304 T^{3} + 5037969 p^{3} T^{4} + 3228 p^{6} T^{5} + p^{9} T^{6} \) |
| 89 | $S_4\times C_2$ | \( 1 + 510 T + 1179207 T^{2} + 945565380 T^{3} + 1179207 p^{3} T^{4} + 510 p^{6} T^{5} + p^{9} T^{6} \) |
| 97 | $S_4\times C_2$ | \( 1 + 2304 T + 3529743 T^{2} + 4206424224 T^{3} + 3529743 p^{3} T^{4} + 2304 p^{6} T^{5} + p^{9} T^{6} \) |
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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
−9.434944619451425174844466565870, −8.646659330971438873308678799518, −8.544182642540491745946145412635, −8.235073914645630647415907045200, −7.85090533276702802823123894245, −7.61688674101837731176852244216, −7.46412340946306662739664228505, −6.80509451747619907227482643667, −6.75062329370279236140532902762, −6.69060322743776989299857845405, −5.99049116302765160198182102350, −5.97440744220727364760899480687, −5.71298660139958639288327519480, −5.16326517845647102646449408998, −4.86837279816890701869839013085, −4.82170042315884956789935957665, −4.03691118041660161029339984508, −3.98838440521622620782906570476, −3.79661218737515057205298799820, −2.95869490276874169124142056231, −2.86324139846877123154526713818, −2.84142368853168686375204251854, −1.29213892719751324525931265547, −1.28461994731532970298288136319, −1.25821445222555334748379021975, 0, 0, 0,
1.25821445222555334748379021975, 1.28461994731532970298288136319, 1.29213892719751324525931265547, 2.84142368853168686375204251854, 2.86324139846877123154526713818, 2.95869490276874169124142056231, 3.79661218737515057205298799820, 3.98838440521622620782906570476, 4.03691118041660161029339984508, 4.82170042315884956789935957665, 4.86837279816890701869839013085, 5.16326517845647102646449408998, 5.71298660139958639288327519480, 5.97440744220727364760899480687, 5.99049116302765160198182102350, 6.69060322743776989299857845405, 6.75062329370279236140532902762, 6.80509451747619907227482643667, 7.46412340946306662739664228505, 7.61688674101837731176852244216, 7.85090533276702802823123894245, 8.235073914645630647415907045200, 8.544182642540491745946145412635, 8.646659330971438873308678799518, 9.434944619451425174844466565870
