| L(s) = 1 | − 3·2-s + 3-s + 6·4-s + 5·5-s − 3·6-s − 7-s − 10·8-s − 3·9-s − 15·10-s + 6·12-s − 5·13-s + 3·14-s + 5·15-s + 15·16-s − 4·17-s + 9·18-s + 3·19-s + 30·20-s − 21-s + 2·23-s − 10·24-s + 7·25-s + 15·26-s − 5·27-s − 6·28-s − 7·29-s − 15·30-s + ⋯ |
| L(s) = 1 | − 2.12·2-s + 0.577·3-s + 3·4-s + 2.23·5-s − 1.22·6-s − 0.377·7-s − 3.53·8-s − 9-s − 4.74·10-s + 1.73·12-s − 1.38·13-s + 0.801·14-s + 1.29·15-s + 15/4·16-s − 0.970·17-s + 2.12·18-s + 0.688·19-s + 6.70·20-s − 0.218·21-s + 0.417·23-s − 2.04·24-s + 7/5·25-s + 2.94·26-s − 0.962·27-s − 1.13·28-s − 1.29·29-s − 2.73·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{3} \cdot 11^{6} \cdot 19^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{3} \cdot 11^{6} \cdot 19^{3}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]
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 | $C_1$ | \( ( 1 + T )^{3} \) |
| 11 | | \( 1 \) |
| 19 | $C_1$ | \( ( 1 - T )^{3} \) |
| good | 3 | $S_4\times C_2$ | \( 1 - T + 4 T^{2} - 2 T^{3} + 4 p T^{4} - p^{2} T^{5} + p^{3} T^{6} \) |
| 5 | $S_4\times C_2$ | \( 1 - p T + 18 T^{2} - 48 T^{3} + 18 p T^{4} - p^{3} T^{5} + p^{3} T^{6} \) |
| 7 | $S_4\times C_2$ | \( 1 + T + 2 p T^{2} + 18 T^{3} + 2 p^{2} T^{4} + p^{2} T^{5} + p^{3} T^{6} \) |
| 13 | $S_4\times C_2$ | \( 1 + 5 T + 40 T^{2} + 116 T^{3} + 40 p T^{4} + 5 p^{2} T^{5} + p^{3} T^{6} \) |
| 17 | $S_4\times C_2$ | \( 1 + 4 T + 27 T^{2} + 48 T^{3} + 27 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 23 | $S_4\times C_2$ | \( 1 - 2 T + 41 T^{2} - 124 T^{3} + 41 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 29 | $S_4\times C_2$ | \( 1 + 7 T + 68 T^{2} + 320 T^{3} + 68 p T^{4} + 7 p^{2} T^{5} + p^{3} T^{6} \) |
| 31 | $S_4\times C_2$ | \( 1 + 3 T + 68 T^{2} + 110 T^{3} + 68 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 37 | $S_4\times C_2$ | \( 1 + 14 T + 127 T^{2} + 908 T^{3} + 127 p T^{4} + 14 p^{2} T^{5} + p^{3} T^{6} \) |
| 41 | $S_4\times C_2$ | \( 1 - 3 T + 98 T^{2} - 268 T^{3} + 98 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 43 | $S_4\times C_2$ | \( 1 - 5 T + 62 T^{2} - 162 T^{3} + 62 p T^{4} - 5 p^{2} T^{5} + p^{3} T^{6} \) |
| 47 | $S_4\times C_2$ | \( 1 + 89 T^{2} + 128 T^{3} + 89 p T^{4} + p^{3} T^{6} \) |
| 53 | $S_4\times C_2$ | \( 1 + 16 T + 223 T^{2} + 1752 T^{3} + 223 p T^{4} + 16 p^{2} T^{5} + p^{3} T^{6} \) |
| 59 | $S_4\times C_2$ | \( 1 + 12 T + 17 T^{2} - 376 T^{3} + 17 p T^{4} + 12 p^{2} T^{5} + p^{3} T^{6} \) |
| 61 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{3} \) |
| 67 | $S_4\times C_2$ | \( 1 - T + 194 T^{2} - 138 T^{3} + 194 p T^{4} - p^{2} T^{5} + p^{3} T^{6} \) |
| 71 | $S_4\times C_2$ | \( 1 + 3 T + 182 T^{2} + 454 T^{3} + 182 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 73 | $S_4\times C_2$ | \( 1 + 4 T + 83 T^{2} + 528 T^{3} + 83 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 79 | $S_4\times C_2$ | \( 1 + 2 T + 121 T^{2} + 668 T^{3} + 121 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 83 | $S_4\times C_2$ | \( 1 + 7 T + 106 T^{2} + 54 T^{3} + 106 p T^{4} + 7 p^{2} T^{5} + p^{3} T^{6} \) |
| 89 | $S_4\times C_2$ | \( 1 - 16 T - 13 T^{2} + 1576 T^{3} - 13 p T^{4} - 16 p^{2} T^{5} + p^{3} T^{6} \) |
| 97 | $S_4\times C_2$ | \( 1 + 14 T + 307 T^{2} + 2588 T^{3} + 307 p T^{4} + 14 p^{2} T^{5} + p^{3} T^{6} \) |
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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
−7.903108339144659327807809891536, −7.44487620750161314651211900116, −7.44479315193929986518296679751, −6.93921448942075696543544491029, −6.89399418351089983991923732378, −6.53130714461499327761721513249, −6.52229166538242133939419240547, −5.87526749231239029101797718274, −5.83979321928814990869847321724, −5.83429289429203503319609857790, −5.37299032170386843663735045648, −5.22184170617061459442466833921, −4.81394445406642176678007515929, −4.64267974149214275071590775314, −4.05791483338553301230678509383, −3.78357062610461840899419784648, −3.19047303599234683116520019577, −3.10554416790168669617721398411, −2.99020902202705458000786845944, −2.48093162165497453002001686116, −2.26390144512762444245296381006, −2.07115015555569128349213722405, −1.71628795460907451386955005869, −1.34966991693543028260195209770, −1.33231216921135747237676385657, 0, 0, 0,
1.33231216921135747237676385657, 1.34966991693543028260195209770, 1.71628795460907451386955005869, 2.07115015555569128349213722405, 2.26390144512762444245296381006, 2.48093162165497453002001686116, 2.99020902202705458000786845944, 3.10554416790168669617721398411, 3.19047303599234683116520019577, 3.78357062610461840899419784648, 4.05791483338553301230678509383, 4.64267974149214275071590775314, 4.81394445406642176678007515929, 5.22184170617061459442466833921, 5.37299032170386843663735045648, 5.83429289429203503319609857790, 5.83979321928814990869847321724, 5.87526749231239029101797718274, 6.52229166538242133939419240547, 6.53130714461499327761721513249, 6.89399418351089983991923732378, 6.93921448942075696543544491029, 7.44479315193929986518296679751, 7.44487620750161314651211900116, 7.903108339144659327807809891536
