| L(s) = 1 | − 3-s − 3·5-s − 5·7-s − 2·9-s − 4·11-s + 5·13-s + 3·15-s − 11·17-s − 3·19-s + 5·21-s − 9·23-s + 6·25-s − 3·27-s + 3·29-s − 14·31-s + 4·33-s + 15·35-s + 14·37-s − 5·39-s − 10·41-s − 10·43-s + 6·45-s + 4·49-s + 11·51-s − 7·53-s + 12·55-s + 3·57-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 1.34·5-s − 1.88·7-s − 2/3·9-s − 1.20·11-s + 1.38·13-s + 0.774·15-s − 2.66·17-s − 0.688·19-s + 1.09·21-s − 1.87·23-s + 6/5·25-s − 0.577·27-s + 0.557·29-s − 2.51·31-s + 0.696·33-s + 2.53·35-s + 2.30·37-s − 0.800·39-s − 1.56·41-s − 1.52·43-s + 0.894·45-s + 4/7·49-s + 1.54·51-s − 0.961·53-s + 1.61·55-s + 0.397·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{9} \cdot 5^{3} \cdot 19^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{9} \cdot 5^{3} \cdot 19^{3}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]
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 | | \( 1 \) |
| 5 | $C_1$ | \( ( 1 + T )^{3} \) |
| 19 | $C_1$ | \( ( 1 + T )^{3} \) |
| good | 3 | $S_4\times C_2$ | \( 1 + T + p T^{2} + 8 T^{3} + p^{2} T^{4} + p^{2} T^{5} + p^{3} T^{6} \) |
| 7 | $S_4\times C_2$ | \( 1 + 5 T + 3 p T^{2} + 62 T^{3} + 3 p^{2} T^{4} + 5 p^{2} T^{5} + p^{3} T^{6} \) |
| 11 | $S_4\times C_2$ | \( 1 + 4 T + 13 T^{2} + 24 T^{3} + 13 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 13 | $S_4\times C_2$ | \( 1 - 5 T + 17 T^{2} - 24 T^{3} + 17 p T^{4} - 5 p^{2} T^{5} + p^{3} T^{6} \) |
| 17 | $S_4\times C_2$ | \( 1 + 11 T + 83 T^{2} + 394 T^{3} + 83 p T^{4} + 11 p^{2} T^{5} + p^{3} T^{6} \) |
| 23 | $S_4\times C_2$ | \( 1 + 9 T + 53 T^{2} + 254 T^{3} + 53 p T^{4} + 9 p^{2} T^{5} + p^{3} T^{6} \) |
| 29 | $S_4\times C_2$ | \( 1 - 3 T + 15 T^{2} - 66 T^{3} + 15 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 31 | $S_4\times C_2$ | \( 1 + 14 T + 125 T^{2} + 804 T^{3} + 125 p T^{4} + 14 p^{2} T^{5} + p^{3} T^{6} \) |
| 37 | $S_4\times C_2$ | \( 1 - 14 T + 157 T^{2} - 1016 T^{3} + 157 p T^{4} - 14 p^{2} T^{5} + p^{3} T^{6} \) |
| 41 | $S_4\times C_2$ | \( 1 + 10 T + 79 T^{2} + 348 T^{3} + 79 p T^{4} + 10 p^{2} T^{5} + p^{3} T^{6} \) |
| 43 | $S_4\times C_2$ | \( 1 + 10 T + 41 T^{2} + 12 T^{3} + 41 p T^{4} + 10 p^{2} T^{5} + p^{3} T^{6} \) |
| 47 | $S_4\times C_2$ | \( 1 + 101 T^{2} + 64 T^{3} + 101 p T^{4} + p^{3} T^{6} \) |
| 53 | $S_4\times C_2$ | \( 1 + 7 T + 145 T^{2} + 744 T^{3} + 145 p T^{4} + 7 p^{2} T^{5} + p^{3} T^{6} \) |
| 59 | $S_4\times C_2$ | \( 1 + 7 T + 185 T^{2} + 818 T^{3} + 185 p T^{4} + 7 p^{2} T^{5} + p^{3} T^{6} \) |
| 61 | $S_4\times C_2$ | \( 1 - 20 T + 215 T^{2} - 1800 T^{3} + 215 p T^{4} - 20 p^{2} T^{5} + p^{3} T^{6} \) |
| 67 | $S_4\times C_2$ | \( 1 + T + p T^{2} + 664 T^{3} + p^{2} T^{4} + p^{2} T^{5} + p^{3} T^{6} \) |
| 71 | $S_4\times C_2$ | \( 1 + 8 T + 133 T^{2} + 624 T^{3} + 133 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 73 | $S_4\times C_2$ | \( 1 + 13 T + 155 T^{2} + 1398 T^{3} + 155 p T^{4} + 13 p^{2} T^{5} + p^{3} T^{6} \) |
| 79 | $S_4\times C_2$ | \( 1 - 14 T + 181 T^{2} - 2228 T^{3} + 181 p T^{4} - 14 p^{2} T^{5} + p^{3} T^{6} \) |
| 83 | $S_4\times C_2$ | \( 1 + 26 T + 441 T^{2} + 4668 T^{3} + 441 p T^{4} + 26 p^{2} T^{5} + p^{3} T^{6} \) |
| 89 | $S_4\times C_2$ | \( 1 - 18 T + 335 T^{2} - 3244 T^{3} + 335 p T^{4} - 18 p^{2} T^{5} + p^{3} T^{6} \) |
| 97 | $S_4\times C_2$ | \( 1 + 6 T + 17 T^{2} - 1144 T^{3} + 17 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) |
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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
−9.765227696294508391884686644925, −9.240599745448566300270306069503, −8.971385524946926485274035941654, −8.883359744290589840987117462222, −8.436146624377183537325106362503, −8.131579358308458661904477506025, −8.055160041239651651367071756563, −7.60657153408733379963511641028, −7.37408657669182395945300157955, −6.77373143567401309148253712603, −6.50290284223620059435927644011, −6.43879339739781149747044737983, −6.38735628882411981497300104999, −5.64800226140176224046869953244, −5.52925259862135634604553574806, −5.28914654442083821422453026470, −4.51705102928421661060724424478, −4.38421611084775338443680824516, −4.00346046542030291818445637280, −3.76015136093424328614137817959, −3.21791826444571074877271097999, −3.15759946591446452613300504056, −2.42865990661798075817824253028, −2.19209759456665165312882172637, −1.50297959141069652117903727409, 0, 0, 0,
1.50297959141069652117903727409, 2.19209759456665165312882172637, 2.42865990661798075817824253028, 3.15759946591446452613300504056, 3.21791826444571074877271097999, 3.76015136093424328614137817959, 4.00346046542030291818445637280, 4.38421611084775338443680824516, 4.51705102928421661060724424478, 5.28914654442083821422453026470, 5.52925259862135634604553574806, 5.64800226140176224046869953244, 6.38735628882411981497300104999, 6.43879339739781149747044737983, 6.50290284223620059435927644011, 6.77373143567401309148253712603, 7.37408657669182395945300157955, 7.60657153408733379963511641028, 8.055160041239651651367071756563, 8.131579358308458661904477506025, 8.436146624377183537325106362503, 8.883359744290589840987117462222, 8.971385524946926485274035941654, 9.240599745448566300270306069503, 9.765227696294508391884686644925
