| L(s) = 1 | − 4·2-s − 9·3-s + 7·4-s − 15·5-s + 36·6-s − 4·7-s − 12·8-s + 54·9-s + 60·10-s + 33·11-s − 63·12-s + 16·14-s + 135·15-s − 7·16-s − 218·17-s − 216·18-s + 146·19-s − 105·20-s + 36·21-s − 132·22-s − 200·23-s + 108·24-s + 150·25-s − 270·27-s − 28·28-s + 68·29-s − 540·30-s + ⋯ |
| L(s) = 1 | − 1.41·2-s − 1.73·3-s + 7/8·4-s − 1.34·5-s + 2.44·6-s − 0.215·7-s − 0.530·8-s + 2·9-s + 1.89·10-s + 0.904·11-s − 1.51·12-s + 0.305·14-s + 2.32·15-s − 0.109·16-s − 3.11·17-s − 2.82·18-s + 1.76·19-s − 1.17·20-s + 0.374·21-s − 1.27·22-s − 1.81·23-s + 0.918·24-s + 6/5·25-s − 1.92·27-s − 0.188·28-s + 0.435·29-s − 3.28·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4492125 ^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4492125 ^{s/2} \, \Gamma_{\C}(s+3/2)^{3} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | $C_1$ | \( ( 1 + p T )^{3} \) |
| 5 | $C_1$ | \( ( 1 + p T )^{3} \) |
| 11 | $C_1$ | \( ( 1 - p T )^{3} \) |
| good | 2 | $S_4\times C_2$ | \( 1 + p^{2} T + 9 T^{2} + 5 p^{2} T^{3} + 9 p^{3} T^{4} + p^{8} T^{5} + p^{9} T^{6} \) |
| 7 | $S_4\times C_2$ | \( 1 + 4 T + 601 T^{2} - 1000 T^{3} + 601 p^{3} T^{4} + 4 p^{6} T^{5} + p^{9} T^{6} \) |
| 13 | $S_4\times C_2$ | \( 1 + 3031 T^{2} - 34144 T^{3} + 3031 p^{3} T^{4} + p^{9} T^{6} \) |
| 17 | $S_4\times C_2$ | \( 1 + 218 T + 24419 T^{2} + 1906964 T^{3} + 24419 p^{3} T^{4} + 218 p^{6} T^{5} + p^{9} T^{6} \) |
| 19 | $S_4\times C_2$ | \( 1 - 146 T + 25953 T^{2} - 2033788 T^{3} + 25953 p^{3} T^{4} - 146 p^{6} T^{5} + p^{9} T^{6} \) |
| 23 | $S_4\times C_2$ | \( 1 + 200 T + 33829 T^{2} + 4868464 T^{3} + 33829 p^{3} T^{4} + 200 p^{6} T^{5} + p^{9} T^{6} \) |
| 29 | $S_4\times C_2$ | \( 1 - 68 T + 18803 T^{2} - 153848 T^{3} + 18803 p^{3} T^{4} - 68 p^{6} T^{5} + p^{9} T^{6} \) |
| 31 | $S_4\times C_2$ | \( 1 + 68 T + 66141 T^{2} + 2239480 T^{3} + 66141 p^{3} T^{4} + 68 p^{6} T^{5} + p^{9} T^{6} \) |
| 37 | $S_4\times C_2$ | \( 1 + 390 T + 188419 T^{2} + 40128292 T^{3} + 188419 p^{3} T^{4} + 390 p^{6} T^{5} + p^{9} T^{6} \) |
| 41 | $S_4\times C_2$ | \( 1 + 196 T + 4407 p T^{2} + 22652824 T^{3} + 4407 p^{4} T^{4} + 196 p^{6} T^{5} + p^{9} T^{6} \) |
| 43 | $S_4\times C_2$ | \( 1 + 524 T + 209853 T^{2} + 52049416 T^{3} + 209853 p^{3} T^{4} + 524 p^{6} T^{5} + p^{9} T^{6} \) |
| 47 | $S_4\times C_2$ | \( 1 + 60 T + 175549 T^{2} - 8508216 T^{3} + 175549 p^{3} T^{4} + 60 p^{6} T^{5} + p^{9} T^{6} \) |
| 53 | $S_4\times C_2$ | \( 1 + 158 T + 185779 T^{2} + 86620084 T^{3} + 185779 p^{3} T^{4} + 158 p^{6} T^{5} + p^{9} T^{6} \) |
| 59 | $S_4\times C_2$ | \( 1 + 1044 T + 680281 T^{2} + 344604088 T^{3} + 680281 p^{3} T^{4} + 1044 p^{6} T^{5} + p^{9} T^{6} \) |
| 61 | $S_4\times C_2$ | \( 1 - 642 T + 620395 T^{2} - 268686220 T^{3} + 620395 p^{3} T^{4} - 642 p^{6} T^{5} + p^{9} T^{6} \) |
| 67 | $S_4\times C_2$ | \( 1 + 236 T + 440081 T^{2} + 229497800 T^{3} + 440081 p^{3} T^{4} + 236 p^{6} T^{5} + p^{9} T^{6} \) |
| 71 | $S_4\times C_2$ | \( 1 + 544 T + 944005 T^{2} + 395960768 T^{3} + 944005 p^{3} T^{4} + 544 p^{6} T^{5} + p^{9} T^{6} \) |
| 73 | $S_4\times C_2$ | \( 1 - 900 T + 867475 T^{2} - 705839944 T^{3} + 867475 p^{3} T^{4} - 900 p^{6} T^{5} + p^{9} T^{6} \) |
| 79 | $S_4\times C_2$ | \( 1 + 1586 T + 2041325 T^{2} + 1549224716 T^{3} + 2041325 p^{3} T^{4} + 1586 p^{6} T^{5} + p^{9} T^{6} \) |
| 83 | $S_4\times C_2$ | \( 1 + 1582 T + 1407717 T^{2} + 884488684 T^{3} + 1407717 p^{3} T^{4} + 1582 p^{6} T^{5} + p^{9} T^{6} \) |
| 89 | $S_4\times C_2$ | \( 1 + 2122 T + 3521847 T^{2} + 3285333068 T^{3} + 3521847 p^{3} T^{4} + 2122 p^{6} T^{5} + p^{9} T^{6} \) |
| 97 | $S_4\times C_2$ | \( 1 - 618 T + 1446319 T^{2} - 1351607564 T^{3} + 1446319 p^{3} T^{4} - 618 p^{6} T^{5} + p^{9} T^{6} \) |
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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
−11.38396107918053199134896155671, −11.31503634546292837979045754117, −11.06426070673374402137520530015, −10.62572089363730959095947307234, −9.987695812572941867857313303660, −9.854746844123005947391193913803, −9.784012941071971075327506411725, −8.883988044396467932812105532638, −8.822250236537784233102160049972, −8.712145943629755070877300170293, −7.901200733731107632500869835234, −7.87615370331142558249828812588, −7.11217076462862682062123649881, −6.82088248415705226338381828771, −6.78837948449091974826726938817, −6.33632740601128452448267112696, −5.88239134962647394570004081477, −5.06730892068729817480269668819, −5.05832509076758630826783281233, −4.23912785895003769931341466511, −4.15210514423586779419605478593, −3.50437855314430610577113437027, −2.79927966384044821933287276777, −1.66260061704225131346789765018, −1.46313092156811801131018048418, 0, 0, 0,
1.46313092156811801131018048418, 1.66260061704225131346789765018, 2.79927966384044821933287276777, 3.50437855314430610577113437027, 4.15210514423586779419605478593, 4.23912785895003769931341466511, 5.05832509076758630826783281233, 5.06730892068729817480269668819, 5.88239134962647394570004081477, 6.33632740601128452448267112696, 6.78837948449091974826726938817, 6.82088248415705226338381828771, 7.11217076462862682062123649881, 7.87615370331142558249828812588, 7.901200733731107632500869835234, 8.712145943629755070877300170293, 8.822250236537784233102160049972, 8.883988044396467932812105532638, 9.784012941071971075327506411725, 9.854746844123005947391193913803, 9.987695812572941867857313303660, 10.62572089363730959095947307234, 11.06426070673374402137520530015, 11.31503634546292837979045754117, 11.38396107918053199134896155671
