L(s) = 1 | − 3·2-s + 3-s + 6·4-s − 2·5-s − 3·6-s − 3·7-s − 10·8-s + 2·9-s + 6·10-s − 7·11-s + 6·12-s + 9·14-s − 2·15-s + 15·16-s + 2·17-s − 6·18-s − 4·19-s − 12·20-s − 3·21-s + 21·22-s + 3·23-s − 10·24-s − 25-s − 5·27-s − 18·28-s + 8·29-s + 6·30-s + ⋯ |
L(s) = 1 | − 2.12·2-s + 0.577·3-s + 3·4-s − 0.894·5-s − 1.22·6-s − 1.13·7-s − 3.53·8-s + 2/3·9-s + 1.89·10-s − 2.11·11-s + 1.73·12-s + 2.40·14-s − 0.516·15-s + 15/4·16-s + 0.485·17-s − 1.41·18-s − 0.917·19-s − 2.68·20-s − 0.654·21-s + 4.47·22-s + 0.625·23-s − 2.04·24-s − 1/5·25-s − 0.962·27-s − 3.40·28-s + 1.48·29-s + 1.09·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{3} \cdot 7^{3} \cdot 13^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{3} \cdot 7^{3} \cdot 13^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.4038082142\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.4038082142\) |
\(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 + T )^{3} \) |
| 7 | $C_1$ | \( ( 1 + T )^{3} \) |
| 13 | | \( 1 \) |
good | 3 | $S_4\times C_2$ | \( 1 - T - T^{2} + 8 T^{3} - p T^{4} - p^{2} T^{5} + p^{3} T^{6} \) |
| 5 | $S_4\times C_2$ | \( 1 + 2 T + p T^{2} + 16 T^{3} + p^{2} T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 11 | $S_4\times C_2$ | \( 1 + 7 T + 25 T^{2} + 6 p T^{3} + 25 p T^{4} + 7 p^{2} T^{5} + p^{3} T^{6} \) |
| 17 | $S_4\times C_2$ | \( 1 - 2 T + 7 T^{2} - 12 T^{3} + 7 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 19 | $S_4\times C_2$ | \( 1 + 4 T + 51 T^{2} + 136 T^{3} + 51 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 23 | $S_4\times C_2$ | \( 1 - 3 T + 53 T^{2} - 106 T^{3} + 53 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 29 | $S_4\times C_2$ | \( 1 - 8 T + 67 T^{2} - 288 T^{3} + 67 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 31 | $S_4\times C_2$ | \( 1 - 7 T + 85 T^{2} - 346 T^{3} + 85 p T^{4} - 7 p^{2} T^{5} + p^{3} T^{6} \) |
| 37 | $S_4\times C_2$ | \( 1 + 9 T + 119 T^{2} + 622 T^{3} + 119 p T^{4} + 9 p^{2} T^{5} + p^{3} T^{6} \) |
| 41 | $S_4\times C_2$ | \( 1 - 15 T + 179 T^{2} - 1274 T^{3} + 179 p T^{4} - 15 p^{2} T^{5} + p^{3} T^{6} \) |
| 43 | $S_4\times C_2$ | \( 1 + 8 T + 105 T^{2} + 560 T^{3} + 105 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 47 | $S_4\times C_2$ | \( 1 + 17 T + 213 T^{2} + 1686 T^{3} + 213 p T^{4} + 17 p^{2} T^{5} + p^{3} T^{6} \) |
| 53 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{3} \) |
| 59 | $S_4\times C_2$ | \( 1 + 10 T + 83 T^{2} + 296 T^{3} + 83 p T^{4} + 10 p^{2} T^{5} + p^{3} T^{6} \) |
| 61 | $S_4\times C_2$ | \( 1 - 5 T + 181 T^{2} - 608 T^{3} + 181 p T^{4} - 5 p^{2} T^{5} + p^{3} T^{6} \) |
| 67 | $S_4\times C_2$ | \( 1 + 15 T + 257 T^{2} + 2026 T^{3} + 257 p T^{4} + 15 p^{2} T^{5} + p^{3} T^{6} \) |
| 71 | $S_4\times C_2$ | \( 1 - 4 T + 53 T^{2} + 328 T^{3} + 53 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 73 | $S_4\times C_2$ | \( 1 - 11 T + 235 T^{2} - 1610 T^{3} + 235 p T^{4} - 11 p^{2} T^{5} + p^{3} T^{6} \) |
| 79 | $S_4\times C_2$ | \( 1 + 7 T + 149 T^{2} + 1254 T^{3} + 149 p T^{4} + 7 p^{2} T^{5} + p^{3} T^{6} \) |
| 83 | $S_4\times C_2$ | \( 1 + 10 T + 123 T^{2} + 408 T^{3} + 123 p T^{4} + 10 p^{2} T^{5} + p^{3} T^{6} \) |
| 89 | $S_4\times C_2$ | \( 1 - 26 T + 447 T^{2} - 4860 T^{3} + 447 p T^{4} - 26 p^{2} T^{5} + p^{3} T^{6} \) |
| 97 | $S_4\times C_2$ | \( 1 - 21 T + 419 T^{2} - 4270 T^{3} + 419 p T^{4} - 21 p^{2} T^{5} + p^{3} T^{6} \) |
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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
−7.966306159794271035592682638386, −7.77196148387797437479645854702, −7.68341207233586439060361828307, −7.40259016720027134605921401170, −7.11477762965912436140015227678, −6.69320041380236837939052127835, −6.63278455346233148793356750805, −6.24221545747172056595185993836, −6.07388852277382194494900203058, −5.86204952224482541970873771375, −5.26133425140622440537315541734, −5.01819187314938525024100768315, −4.76828818688048193587200068964, −4.46789743353455434640982989969, −3.96107169174574519624129524081, −3.67751743626183808932281260047, −3.12054334281889606515934889533, −3.06149081863010575768760655536, −2.98174395720941023302449127133, −2.45959761493621796936091212325, −2.05897451687618667388423324755, −1.71988958744105817278795198472, −1.33183753168464819506923358904, −0.47838941918653432089481264991, −0.37494876361692562821789927679,
0.37494876361692562821789927679, 0.47838941918653432089481264991, 1.33183753168464819506923358904, 1.71988958744105817278795198472, 2.05897451687618667388423324755, 2.45959761493621796936091212325, 2.98174395720941023302449127133, 3.06149081863010575768760655536, 3.12054334281889606515934889533, 3.67751743626183808932281260047, 3.96107169174574519624129524081, 4.46789743353455434640982989969, 4.76828818688048193587200068964, 5.01819187314938525024100768315, 5.26133425140622440537315541734, 5.86204952224482541970873771375, 6.07388852277382194494900203058, 6.24221545747172056595185993836, 6.63278455346233148793356750805, 6.69320041380236837939052127835, 7.11477762965912436140015227678, 7.40259016720027134605921401170, 7.68341207233586439060361828307, 7.77196148387797437479645854702, 7.966306159794271035592682638386