| L(s) = 1 | − 2·3-s − 2·7-s − 11-s − 3·13-s − 6·17-s + 3·19-s + 4·21-s − 16·23-s + 5·27-s + 3·29-s − 31-s + 2·33-s + 13·37-s + 6·39-s − 3·41-s − 13·43-s − 9·47-s + 2·49-s + 12·51-s + 53-s − 6·57-s − 6·59-s − 5·61-s − 19·67-s + 32·69-s − 3·71-s − 15·73-s + ⋯ |
| L(s) = 1 | − 1.15·3-s − 0.755·7-s − 0.301·11-s − 0.832·13-s − 1.45·17-s + 0.688·19-s + 0.872·21-s − 3.33·23-s + 0.962·27-s + 0.557·29-s − 0.179·31-s + 0.348·33-s + 2.13·37-s + 0.960·39-s − 0.468·41-s − 1.98·43-s − 1.31·47-s + 2/7·49-s + 1.68·51-s + 0.137·53-s − 0.794·57-s − 0.781·59-s − 0.640·61-s − 2.32·67-s + 3.85·69-s − 0.356·71-s − 1.75·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{6} \cdot 5^{6} \cdot 19^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{6} \cdot 5^{6} \cdot 19^{3}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]
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 | | \( 1 \) |
| 5 | | \( 1 \) |
| 19 | $C_1$ | \( ( 1 - T )^{3} \) |
| good | 3 | $D_{6}$ | \( 1 + 2 T + 4 T^{2} + p T^{3} + 4 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 7 | $D_{6}$ | \( 1 + 2 T + 2 T^{2} - 19 T^{3} + 2 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 11 | $S_4\times C_2$ | \( 1 + T + 27 T^{2} + 25 T^{3} + 27 p T^{4} + p^{2} T^{5} + p^{3} T^{6} \) |
| 13 | $S_4\times C_2$ | \( 1 + 3 T + 33 T^{2} + 71 T^{3} + 33 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 17 | $S_4\times C_2$ | \( 1 + 6 T + 36 T^{2} + 105 T^{3} + 36 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 23 | $S_4\times C_2$ | \( 1 + 16 T + 150 T^{2} + 865 T^{3} + 150 p T^{4} + 16 p^{2} T^{5} + p^{3} T^{6} \) |
| 29 | $S_4\times C_2$ | \( 1 - 3 T + 63 T^{2} - 201 T^{3} + 63 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 31 | $S_4\times C_2$ | \( 1 + T + 53 T^{2} - 47 T^{3} + 53 p T^{4} + p^{2} T^{5} + p^{3} T^{6} \) |
| 37 | $S_4\times C_2$ | \( 1 - 13 T + 161 T^{2} - 1021 T^{3} + 161 p T^{4} - 13 p^{2} T^{5} + p^{3} T^{6} \) |
| 41 | $S_4\times C_2$ | \( 1 + 3 T + 87 T^{2} + 219 T^{3} + 87 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 43 | $S_4\times C_2$ | \( 1 + 13 T + 160 T^{2} + 1049 T^{3} + 160 p T^{4} + 13 p^{2} T^{5} + p^{3} T^{6} \) |
| 47 | $S_4\times C_2$ | \( 1 + 9 T + 45 T^{2} + 225 T^{3} + 45 p T^{4} + 9 p^{2} T^{5} + p^{3} T^{6} \) |
| 53 | $S_4\times C_2$ | \( 1 - T + 90 T^{2} - 109 T^{3} + 90 p T^{4} - p^{2} T^{5} + p^{3} T^{6} \) |
| 59 | $S_4\times C_2$ | \( 1 + 6 T + 153 T^{2} + 636 T^{3} + 153 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 61 | $S_4\times C_2$ | \( 1 + 5 T + 85 T^{2} + 121 T^{3} + 85 p T^{4} + 5 p^{2} T^{5} + p^{3} T^{6} \) |
| 67 | $S_4\times C_2$ | \( 1 + 19 T + 175 T^{2} + 1223 T^{3} + 175 p T^{4} + 19 p^{2} T^{5} + p^{3} T^{6} \) |
| 71 | $S_4\times C_2$ | \( 1 + 3 T + 180 T^{2} + 399 T^{3} + 180 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 73 | $S_4\times C_2$ | \( 1 + 15 T + 261 T^{2} + 2141 T^{3} + 261 p T^{4} + 15 p^{2} T^{5} + p^{3} T^{6} \) |
| 79 | $S_4\times C_2$ | \( 1 - 2 T + 80 T^{2} - 845 T^{3} + 80 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 83 | $S_4\times C_2$ | \( 1 + 29 T + 525 T^{2} + 5675 T^{3} + 525 p T^{4} + 29 p^{2} T^{5} + p^{3} T^{6} \) |
| 89 | $S_4\times C_2$ | \( 1 - 14 T + 258 T^{2} - 2003 T^{3} + 258 p T^{4} - 14 p^{2} T^{5} + p^{3} T^{6} \) |
| 97 | $S_4\times C_2$ | \( 1 - T + 137 T^{2} - 877 T^{3} + 137 p T^{4} - p^{2} T^{5} + p^{3} T^{6} \) |
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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
−8.558813969526458143757678756802, −8.173881184039491223135271978439, −8.048985526449180672080670061654, −7.85994217001762863168567379537, −7.48039322145522093271554322005, −7.19114649932939328307252874453, −6.98150973979863020460703349302, −6.43873843379510083771439849346, −6.38964635051885969857483646673, −6.29357708955303672461640857227, −5.80232781463336422858395235642, −5.77246462337822238244804741764, −5.44617507963906731649615003355, −4.90639725851899570369214566549, −4.83357086259240280427797320736, −4.46532343682410600224227115559, −4.20231458858107885808251515284, −3.94046108998624592801582996816, −3.53063917703608566567030418224, −2.89571547232867225181889414913, −2.88780524972653745712949156724, −2.56063613740866609124257953785, −2.01822135036020651751822802961, −1.58895211039682914820742805029, −1.24315191819665381359586901850, 0, 0, 0,
1.24315191819665381359586901850, 1.58895211039682914820742805029, 2.01822135036020651751822802961, 2.56063613740866609124257953785, 2.88780524972653745712949156724, 2.89571547232867225181889414913, 3.53063917703608566567030418224, 3.94046108998624592801582996816, 4.20231458858107885808251515284, 4.46532343682410600224227115559, 4.83357086259240280427797320736, 4.90639725851899570369214566549, 5.44617507963906731649615003355, 5.77246462337822238244804741764, 5.80232781463336422858395235642, 6.29357708955303672461640857227, 6.38964635051885969857483646673, 6.43873843379510083771439849346, 6.98150973979863020460703349302, 7.19114649932939328307252874453, 7.48039322145522093271554322005, 7.85994217001762863168567379537, 8.048985526449180672080670061654, 8.173881184039491223135271978439, 8.558813969526458143757678756802
