| L(s) = 1 | + 2-s − 3·4-s + 6·5-s − 4·8-s + 6·10-s − 11-s − 4·13-s + 3·16-s − 18·19-s − 18·20-s − 22-s − 7·23-s + 9·25-s − 4·26-s − 5·29-s − 8·31-s + 6·32-s − 19·37-s − 18·38-s − 24·40-s + 2·41-s + 43-s + 3·44-s − 7·46-s + 18·47-s + 9·50-s + 12·52-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 3/2·4-s + 2.68·5-s − 1.41·8-s + 1.89·10-s − 0.301·11-s − 1.10·13-s + 3/4·16-s − 4.12·19-s − 4.02·20-s − 0.213·22-s − 1.45·23-s + 9/5·25-s − 0.784·26-s − 0.928·29-s − 1.43·31-s + 1.06·32-s − 3.12·37-s − 2.91·38-s − 3.79·40-s + 0.312·41-s + 0.152·43-s + 0.452·44-s − 1.03·46-s + 2.62·47-s + 1.27·50-s + 1.66·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{6} \cdot 7^{9}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{6} \cdot 7^{9}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]
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 | 3 | | \( 1 \) |
| 7 | | \( 1 \) |
| good | 2 | $A_4\times C_2$ | \( 1 - T + p^{2} T^{2} - 3 T^{3} + p^{3} T^{4} - p^{2} T^{5} + p^{3} T^{6} \) |
| 5 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{3} \) |
| 11 | $A_4\times C_2$ | \( 1 + T + 31 T^{2} + 21 T^{3} + 31 p T^{4} + p^{2} T^{5} + p^{3} T^{6} \) |
| 13 | $A_4\times C_2$ | \( 1 + 4 T + 35 T^{2} + 96 T^{3} + 35 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 17 | $A_4\times C_2$ | \( 1 + 23 T^{2} - 56 T^{3} + 23 p T^{4} + p^{3} T^{6} \) |
| 19 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{3} \) |
| 23 | $A_4\times C_2$ | \( 1 + 7 T + 41 T^{2} + 119 T^{3} + 41 p T^{4} + 7 p^{2} T^{5} + p^{3} T^{6} \) |
| 29 | $A_4\times C_2$ | \( 1 + 5 T + 93 T^{2} + 291 T^{3} + 93 p T^{4} + 5 p^{2} T^{5} + p^{3} T^{6} \) |
| 31 | $A_4\times C_2$ | \( 1 + 8 T + 49 T^{2} + 152 T^{3} + 49 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 37 | $A_4\times C_2$ | \( 1 + 19 T + 201 T^{2} + 1435 T^{3} + 201 p T^{4} + 19 p^{2} T^{5} + p^{3} T^{6} \) |
| 41 | $A_4\times C_2$ | \( 1 - 2 T + 59 T^{2} + 68 T^{3} + 59 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 43 | $A_4\times C_2$ | \( 1 - T + 57 T^{2} - 127 T^{3} + 57 p T^{4} - p^{2} T^{5} + p^{3} T^{6} \) |
| 47 | $A_4\times C_2$ | \( 1 - 18 T + 221 T^{2} - 1796 T^{3} + 221 p T^{4} - 18 p^{2} T^{5} + p^{3} T^{6} \) |
| 53 | $A_4\times C_2$ | \( 1 + T + 115 T^{2} + 189 T^{3} + 115 p T^{4} + p^{2} T^{5} + p^{3} T^{6} \) |
| 59 | $A_4\times C_2$ | \( 1 - 22 T + 301 T^{2} - 2700 T^{3} + 301 p T^{4} - 22 p^{2} T^{5} + p^{3} T^{6} \) |
| 61 | $A_4\times C_2$ | \( 1 + 12 T + 119 T^{2} + 632 T^{3} + 119 p T^{4} + 12 p^{2} T^{5} + p^{3} T^{6} \) |
| 67 | $A_4\times C_2$ | \( 1 + 15 T + 269 T^{2} + 2093 T^{3} + 269 p T^{4} + 15 p^{2} T^{5} + p^{3} T^{6} \) |
| 71 | $A_4\times C_2$ | \( 1 + 17 T + 223 T^{2} + 2415 T^{3} + 223 p T^{4} + 17 p^{2} T^{5} + p^{3} T^{6} \) |
| 73 | $A_4\times C_2$ | \( 1 + 16 T + 239 T^{2} + 1992 T^{3} + 239 p T^{4} + 16 p^{2} T^{5} + p^{3} T^{6} \) |
| 79 | $A_4\times C_2$ | \( 1 - 11 T + 261 T^{2} - 1709 T^{3} + 261 p T^{4} - 11 p^{2} T^{5} + p^{3} T^{6} \) |
| 83 | $A_4\times C_2$ | \( 1 - 24 T + 413 T^{2} - 4216 T^{3} + 413 p T^{4} - 24 p^{2} T^{5} + p^{3} T^{6} \) |
| 89 | $A_4\times C_2$ | \( 1 + 6 T + 83 T^{2} + 292 T^{3} + 83 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 97 | $A_4\times C_2$ | \( 1 + 4 T + 119 T^{2} + 1440 T^{3} + 119 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \) |
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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
−8.390120989498286185401076721750, −7.75987917835222769288494757095, −7.60429383538391779247866604613, −7.31567183391396743379966869897, −7.00395174497938949798657969984, −6.62120387042006496107877863704, −6.51174722408501183766724741303, −6.00281541613503069565060553481, −5.91474108954457878065389761052, −5.89320050740295061333001514707, −5.43485633940292560278441215361, −5.28921009875620060265390476922, −5.20056470209423979851985257923, −4.50767001848762291626256690473, −4.43273043885202431453111698568, −4.39913719096381879994293015194, −3.75770529070380217346444127638, −3.70985242578209220456635164258, −3.57833540478086527453349331561, −2.59587732501824488896408213277, −2.44595345389218832571345731479, −2.18458099564218129198177400467, −2.15029840838319665926459337895, −1.56337890784849716216942096282, −1.46629063618514410801234760090, 0, 0, 0,
1.46629063618514410801234760090, 1.56337890784849716216942096282, 2.15029840838319665926459337895, 2.18458099564218129198177400467, 2.44595345389218832571345731479, 2.59587732501824488896408213277, 3.57833540478086527453349331561, 3.70985242578209220456635164258, 3.75770529070380217346444127638, 4.39913719096381879994293015194, 4.43273043885202431453111698568, 4.50767001848762291626256690473, 5.20056470209423979851985257923, 5.28921009875620060265390476922, 5.43485633940292560278441215361, 5.89320050740295061333001514707, 5.91474108954457878065389761052, 6.00281541613503069565060553481, 6.51174722408501183766724741303, 6.62120387042006496107877863704, 7.00395174497938949798657969984, 7.31567183391396743379966869897, 7.60429383538391779247866604613, 7.75987917835222769288494757095, 8.390120989498286185401076721750
