| L(s) = 1 | + 3-s − 2·5-s + 3·7-s + 9-s − 7·11-s − 2·15-s + 17-s + 3·19-s + 3·21-s − 16·23-s − 2·25-s − 5·27-s + 29-s + 5·31-s − 7·33-s − 6·35-s + 6·37-s + 41-s + 4·43-s − 2·45-s + 14·47-s + 6·49-s + 51-s − 31·53-s + 14·55-s + 3·57-s − 18·59-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.894·5-s + 1.13·7-s + 1/3·9-s − 2.11·11-s − 0.516·15-s + 0.242·17-s + 0.688·19-s + 0.654·21-s − 3.33·23-s − 2/5·25-s − 0.962·27-s + 0.185·29-s + 0.898·31-s − 1.21·33-s − 1.01·35-s + 0.986·37-s + 0.156·41-s + 0.609·43-s − 0.298·45-s + 2.04·47-s + 6/7·49-s + 0.140·51-s − 4.25·53-s + 1.88·55-s + 0.397·57-s − 2.34·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{18} \cdot 7^{3} \cdot 19^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{18} \cdot 7^{3} \cdot 19^{3}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]
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 | | \( 1 \) |
| 7 | $C_1$ | \( ( 1 - T )^{3} \) |
| 19 | $C_1$ | \( ( 1 - T )^{3} \) |
| good | 3 | $S_4\times C_2$ | \( 1 - T + 2 p T^{3} - p^{2} T^{5} + p^{3} T^{6} \) |
| 5 | $S_4\times C_2$ | \( 1 + 2 T + 6 T^{2} + 14 T^{3} + 6 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 11 | $S_4\times C_2$ | \( 1 + 7 T + 34 T^{2} + 118 T^{3} + 34 p T^{4} + 7 p^{2} T^{5} + p^{3} T^{6} \) |
| 13 | $D_{6}$ | \( 1 - 54 T^{3} + p^{3} T^{6} \) |
| 17 | $S_4\times C_2$ | \( 1 - T + 36 T^{2} - 16 T^{3} + 36 p T^{4} - p^{2} T^{5} + p^{3} T^{6} \) |
| 23 | $S_4\times C_2$ | \( 1 + 16 T + 144 T^{2} + 832 T^{3} + 144 p T^{4} + 16 p^{2} T^{5} + p^{3} T^{6} \) |
| 29 | $S_4\times C_2$ | \( 1 - T + 72 T^{2} - 40 T^{3} + 72 p T^{4} - p^{2} T^{5} + p^{3} T^{6} \) |
| 31 | $S_4\times C_2$ | \( 1 - 5 T + 86 T^{2} - 302 T^{3} + 86 p T^{4} - 5 p^{2} T^{5} + p^{3} T^{6} \) |
| 37 | $S_4\times C_2$ | \( 1 - 6 T + 30 T^{2} - 282 T^{3} + 30 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 41 | $S_4\times C_2$ | \( 1 - T + 42 T^{2} + 200 T^{3} + 42 p T^{4} - p^{2} T^{5} + p^{3} T^{6} \) |
| 43 | $S_4\times C_2$ | \( 1 - 4 T + 93 T^{2} - 296 T^{3} + 93 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 47 | $S_4\times C_2$ | \( 1 - 14 T + 150 T^{2} - 1100 T^{3} + 150 p T^{4} - 14 p^{2} T^{5} + p^{3} T^{6} \) |
| 53 | $S_4\times C_2$ | \( 1 + 31 T + 470 T^{2} + 4284 T^{3} + 470 p T^{4} + 31 p^{2} T^{5} + p^{3} T^{6} \) |
| 59 | $S_4\times C_2$ | \( 1 + 18 T + 246 T^{2} + 2160 T^{3} + 246 p T^{4} + 18 p^{2} T^{5} + p^{3} T^{6} \) |
| 61 | $S_4\times C_2$ | \( 1 + 26 T + 398 T^{2} + 3734 T^{3} + 398 p T^{4} + 26 p^{2} T^{5} + p^{3} T^{6} \) |
| 67 | $S_4\times C_2$ | \( 1 - T + 94 T^{2} - 110 T^{3} + 94 p T^{4} - p^{2} T^{5} + p^{3} T^{6} \) |
| 71 | $S_4\times C_2$ | \( 1 + 12 T + 168 T^{2} + 1380 T^{3} + 168 p T^{4} + 12 p^{2} T^{5} + p^{3} T^{6} \) |
| 73 | $S_4\times C_2$ | \( 1 - 19 T + 330 T^{2} - 2960 T^{3} + 330 p T^{4} - 19 p^{2} T^{5} + p^{3} T^{6} \) |
| 79 | $S_4\times C_2$ | \( 1 - 4 T + 201 T^{2} - 584 T^{3} + 201 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 83 | $S_4\times C_2$ | \( 1 - 13 T + 4 T^{2} + 1070 T^{3} + 4 p T^{4} - 13 p^{2} T^{5} + p^{3} T^{6} \) |
| 89 | $S_4\times C_2$ | \( 1 + 12 T + 147 T^{2} + 704 T^{3} + 147 p T^{4} + 12 p^{2} T^{5} + p^{3} T^{6} \) |
| 97 | $S_4\times C_2$ | \( 1 + 8 T + 156 T^{2} + 1906 T^{3} + 156 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} \) |
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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
−7.56166279582442477877951290345, −7.31124340262961857695403791835, −6.69375115936026394808955346389, −6.45960544027304772917685919412, −6.17972968032011986455326086597, −6.09031163492901326334643082658, −5.95168812795559649084778501366, −5.49085967809281739481999438835, −5.39260630696215541654706047723, −5.04974659806428064894944946080, −4.84694033177216992738914313969, −4.62162763805082393144840770843, −4.33255549118268647861889763191, −4.06792939439328067284332581115, −3.89038482177280902377693877958, −3.85494620602601227809270512194, −3.26931899823699679777723016490, −2.87033798465278897083328579224, −2.83374658438719505969266525247, −2.69883093044718487810830946938, −2.18215707412620377421419462156, −1.89549392640071417997303602117, −1.69506803068032357291067361311, −1.33328031139999342752319361856, −0.995622444523982185738811758861, 0, 0, 0,
0.995622444523982185738811758861, 1.33328031139999342752319361856, 1.69506803068032357291067361311, 1.89549392640071417997303602117, 2.18215707412620377421419462156, 2.69883093044718487810830946938, 2.83374658438719505969266525247, 2.87033798465278897083328579224, 3.26931899823699679777723016490, 3.85494620602601227809270512194, 3.89038482177280902377693877958, 4.06792939439328067284332581115, 4.33255549118268647861889763191, 4.62162763805082393144840770843, 4.84694033177216992738914313969, 5.04974659806428064894944946080, 5.39260630696215541654706047723, 5.49085967809281739481999438835, 5.95168812795559649084778501366, 6.09031163492901326334643082658, 6.17972968032011986455326086597, 6.45960544027304772917685919412, 6.69375115936026394808955346389, 7.31124340262961857695403791835, 7.56166279582442477877951290345
