| L(s) = 1 | − 6·5-s + 3·7-s + 3·11-s − 3·13-s − 9·17-s + 3·19-s + 6·23-s + 12·25-s − 12·29-s + 12·31-s − 18·35-s − 3·37-s + 3·41-s + 12·43-s − 6·47-s − 6·49-s − 18·53-s − 18·55-s − 21·59-s + 6·61-s + 18·65-s − 6·67-s − 9·71-s + 6·73-s + 9·77-s − 6·79-s − 6·83-s + ⋯ |
| L(s) = 1 | − 2.68·5-s + 1.13·7-s + 0.904·11-s − 0.832·13-s − 2.18·17-s + 0.688·19-s + 1.25·23-s + 12/5·25-s − 2.22·29-s + 2.15·31-s − 3.04·35-s − 0.493·37-s + 0.468·41-s + 1.82·43-s − 0.875·47-s − 6/7·49-s − 2.47·53-s − 2.42·55-s − 2.73·59-s + 0.768·61-s + 2.23·65-s − 0.733·67-s − 1.06·71-s + 0.702·73-s + 1.02·77-s − 0.675·79-s − 0.658·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{15}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{15}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]
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 | | \( 1 \) |
| 3 | | \( 1 \) |
| good | 5 | $A_4\times C_2$ | \( 1 + 6 T + 24 T^{2} + 63 T^{3} + 24 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 7 | $A_4\times C_2$ | \( 1 - 3 T + 15 T^{2} - 25 T^{3} + 15 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 11 | $A_4\times C_2$ | \( 1 - 3 T + 15 T^{2} - 63 T^{3} + 15 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 13 | $A_4\times C_2$ | \( 1 + 3 T + 33 T^{2} + 61 T^{3} + 33 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 17 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{3} \) |
| 19 | $A_4\times C_2$ | \( 1 - 3 T + 33 T^{2} - 115 T^{3} + 33 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 23 | $A_4\times C_2$ | \( 1 - 6 T + 60 T^{2} - 225 T^{3} + 60 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 29 | $A_4\times C_2$ | \( 1 + 12 T + 114 T^{2} + 639 T^{3} + 114 p T^{4} + 12 p^{2} T^{5} + p^{3} T^{6} \) |
| 31 | $A_4\times C_2$ | \( 1 - 12 T + 132 T^{2} - 763 T^{3} + 132 p T^{4} - 12 p^{2} T^{5} + p^{3} T^{6} \) |
| 37 | $A_4\times C_2$ | \( 1 + 3 T + 87 T^{2} + 223 T^{3} + 87 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 41 | $A_4\times C_2$ | \( 1 - 3 T + 69 T^{2} - 27 T^{3} + 69 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 43 | $A_4\times C_2$ | \( 1 - 12 T + 168 T^{2} - 1051 T^{3} + 168 p T^{4} - 12 p^{2} T^{5} + p^{3} T^{6} \) |
| 47 | $A_4\times C_2$ | \( 1 + 6 T + 78 T^{2} + 297 T^{3} + 78 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 53 | $A_4\times C_2$ | \( 1 + 18 T + 240 T^{2} + 1989 T^{3} + 240 p T^{4} + 18 p^{2} T^{5} + p^{3} T^{6} \) |
| 59 | $A_4\times C_2$ | \( 1 + 21 T + 321 T^{2} + 2799 T^{3} + 321 p T^{4} + 21 p^{2} T^{5} + p^{3} T^{6} \) |
| 61 | $A_4\times C_2$ | \( 1 - 6 T + 132 T^{2} - 785 T^{3} + 132 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 67 | $A_4\times C_2$ | \( 1 + 6 T + 150 T^{2} + 695 T^{3} + 150 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 71 | $A_4\times C_2$ | \( 1 + 9 T + 51 T^{2} + 279 T^{3} + 51 p T^{4} + 9 p^{2} T^{5} + p^{3} T^{6} \) |
| 73 | $A_4\times C_2$ | \( 1 - 6 T + 150 T^{2} - 479 T^{3} + 150 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 79 | $A_4\times C_2$ | \( 1 + 6 T + 186 T^{2} + 1001 T^{3} + 186 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 83 | $A_4\times C_2$ | \( 1 + 6 T + 222 T^{2} + 945 T^{3} + 222 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 89 | $A_4\times C_2$ | \( 1 + 78 T^{2} - 999 T^{3} + 78 p T^{4} + p^{3} T^{6} \) |
| 97 | $A_4\times C_2$ | \( 1 - 15 T + 222 T^{2} - 2891 T^{3} + 222 p T^{4} - 15 p^{2} T^{5} + p^{3} T^{6} \) |
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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
−7.949006234235659012259460467551, −7.61900457199935498004591451961, −7.47121996593214277545433940460, −7.36901086495246683706311938885, −6.85504095659697584647232344179, −6.73255519659357397100650458680, −6.62881203407608945484571557486, −6.10289652796523199982805761756, −5.89022826964615820466444961335, −5.77314750969281902168893971711, −4.99473379624509408119723929810, −4.85510581554273203279306503146, −4.83341215912606419559855530729, −4.62926178415234484386524268833, −4.23953675827492579621695492413, −4.10319146589635572577495966231, −3.65736091024836392054639909659, −3.57230502601571549533672351001, −3.37191862324751162921187324734, −2.62758414343559060131625559676, −2.57177888177779657641788038413, −2.41520092911475986425798225864, −1.53061570986950636523135884593, −1.30793952740054271621150881334, −1.28298563183084351914027469446, 0, 0, 0,
1.28298563183084351914027469446, 1.30793952740054271621150881334, 1.53061570986950636523135884593, 2.41520092911475986425798225864, 2.57177888177779657641788038413, 2.62758414343559060131625559676, 3.37191862324751162921187324734, 3.57230502601571549533672351001, 3.65736091024836392054639909659, 4.10319146589635572577495966231, 4.23953675827492579621695492413, 4.62926178415234484386524268833, 4.83341215912606419559855530729, 4.85510581554273203279306503146, 4.99473379624509408119723929810, 5.77314750969281902168893971711, 5.89022826964615820466444961335, 6.10289652796523199982805761756, 6.62881203407608945484571557486, 6.73255519659357397100650458680, 6.85504095659697584647232344179, 7.36901086495246683706311938885, 7.47121996593214277545433940460, 7.61900457199935498004591451961, 7.949006234235659012259460467551
