| L(s) = 1 | − 3·2-s + 3·4-s + 3·7-s + 3·11-s + 3·13-s − 9·14-s − 3·16-s − 9·17-s − 3·19-s − 9·22-s − 6·23-s − 9·26-s + 9·28-s + 12·29-s − 12·31-s + 6·32-s + 27·34-s + 3·37-s + 9·38-s − 3·41-s + 12·43-s + 9·44-s + 18·46-s + 6·47-s − 6·49-s + 9·52-s − 18·53-s + ⋯ |
| L(s) = 1 | − 2.12·2-s + 3/2·4-s + 1.13·7-s + 0.904·11-s + 0.832·13-s − 2.40·14-s − 3/4·16-s − 2.18·17-s − 0.688·19-s − 1.91·22-s − 1.25·23-s − 1.76·26-s + 1.70·28-s + 2.22·29-s − 2.15·31-s + 1.06·32-s + 4.63·34-s + 0.493·37-s + 1.45·38-s − 0.468·41-s + 1.82·43-s + 1.35·44-s + 2.65·46-s + 0.875·47-s − 6/7·49-s + 1.24·52-s − 2.47·53-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{15} \cdot 5^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{15} \cdot 5^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]
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 | 3 | | \( 1 \) |
| 5 | | \( 1 \) |
| good | 2 | $A_4\times C_2$ | \( 1 + 3 T + 3 p T^{2} + 9 T^{3} + 3 p^{2} T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 7 | $A_4\times C_2$ | \( 1 - 3 T + 15 T^{2} - 25 T^{3} + 15 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 11 | $A_4\times C_2$ | \( 1 - 3 T + 15 T^{2} - 63 T^{3} + 15 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 13 | $A_4\times C_2$ | \( 1 - 3 T + 33 T^{2} - 61 T^{3} + 33 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 17 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{3} \) |
| 19 | $A_4\times C_2$ | \( 1 + 3 T + 33 T^{2} + 115 T^{3} + 33 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 23 | $A_4\times C_2$ | \( 1 + 6 T + 60 T^{2} + 225 T^{3} + 60 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 29 | $A_4\times C_2$ | \( 1 - 12 T + 114 T^{2} - 639 T^{3} + 114 p T^{4} - 12 p^{2} T^{5} + p^{3} T^{6} \) |
| 31 | $A_4\times C_2$ | \( 1 + 12 T + 132 T^{2} + 763 T^{3} + 132 p T^{4} + 12 p^{2} T^{5} + p^{3} T^{6} \) |
| 37 | $A_4\times C_2$ | \( 1 - 3 T + 87 T^{2} - 223 T^{3} + 87 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 41 | $A_4\times C_2$ | \( 1 + 3 T + 69 T^{2} + 27 T^{3} + 69 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 43 | $A_4\times C_2$ | \( 1 - 12 T + 168 T^{2} - 1051 T^{3} + 168 p T^{4} - 12 p^{2} T^{5} + p^{3} T^{6} \) |
| 47 | $A_4\times C_2$ | \( 1 - 6 T + 78 T^{2} - 297 T^{3} + 78 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 53 | $A_4\times C_2$ | \( 1 + 18 T + 240 T^{2} + 1989 T^{3} + 240 p T^{4} + 18 p^{2} T^{5} + p^{3} T^{6} \) |
| 59 | $A_4\times C_2$ | \( 1 + 21 T + 321 T^{2} + 2799 T^{3} + 321 p T^{4} + 21 p^{2} T^{5} + p^{3} T^{6} \) |
| 61 | $A_4\times C_2$ | \( 1 - 6 T + 132 T^{2} - 785 T^{3} + 132 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 67 | $A_4\times C_2$ | \( 1 + 6 T + 150 T^{2} + 695 T^{3} + 150 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 71 | $A_4\times C_2$ | \( 1 + 9 T + 51 T^{2} + 279 T^{3} + 51 p T^{4} + 9 p^{2} T^{5} + p^{3} T^{6} \) |
| 73 | $A_4\times C_2$ | \( 1 + 6 T + 150 T^{2} + 479 T^{3} + 150 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 79 | $A_4\times C_2$ | \( 1 - 6 T + 186 T^{2} - 1001 T^{3} + 186 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 83 | $A_4\times C_2$ | \( 1 - 6 T + 222 T^{2} - 945 T^{3} + 222 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 89 | $A_4\times C_2$ | \( 1 + 78 T^{2} + 999 T^{3} + 78 p T^{4} + p^{3} T^{6} \) |
| 97 | $A_4\times C_2$ | \( 1 + 15 T + 222 T^{2} + 2891 T^{3} + 222 p T^{4} + 15 p^{2} T^{5} + p^{3} T^{6} \) |
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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
−7.84804913572489590067511733674, −7.42621973200432391385946500807, −6.94961618739918637285330960384, −6.86218333887521353768846087741, −6.58645730362305916884749551215, −6.54690798209366114558386883613, −6.12911762025219522859363534607, −5.86106486861688125331379647416, −5.82648573387113875464321016358, −5.39370653804434718532267595459, −4.92996052654696826102932867699, −4.72177939738493127875220666949, −4.56723566526087356439765405607, −4.30254031411426570388249375963, −4.06746536357856589901577108100, −3.99063031822420065825602314096, −3.31652000597095841766953321772, −3.30018034950314184729585944020, −2.68114701243556863314160253633, −2.49634212765826383061684132209, −2.01669793715237196736095951242, −1.98062722212635184483077264512, −1.35665500303308230453314082239, −1.19938134173462756685709526805, −1.14905064112869531691792171412, 0, 0, 0,
1.14905064112869531691792171412, 1.19938134173462756685709526805, 1.35665500303308230453314082239, 1.98062722212635184483077264512, 2.01669793715237196736095951242, 2.49634212765826383061684132209, 2.68114701243556863314160253633, 3.30018034950314184729585944020, 3.31652000597095841766953321772, 3.99063031822420065825602314096, 4.06746536357856589901577108100, 4.30254031411426570388249375963, 4.56723566526087356439765405607, 4.72177939738493127875220666949, 4.92996052654696826102932867699, 5.39370653804434718532267595459, 5.82648573387113875464321016358, 5.86106486861688125331379647416, 6.12911762025219522859363534607, 6.54690798209366114558386883613, 6.58645730362305916884749551215, 6.86218333887521353768846087741, 6.94961618739918637285330960384, 7.42621973200432391385946500807, 7.84804913572489590067511733674
