| L(s) = 1 | − 2-s − 4-s − 2·5-s − 3·7-s + 8-s + 2·10-s − 2·11-s + 3·13-s + 3·14-s − 16-s − 4·17-s − 4·19-s + 2·20-s + 2·22-s − 10·23-s − 8·25-s − 3·26-s + 3·28-s − 24·29-s − 4·31-s + 32-s + 4·34-s + 6·35-s + 4·38-s − 2·40-s − 2·41-s + 10·43-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1/2·4-s − 0.894·5-s − 1.13·7-s + 0.353·8-s + 0.632·10-s − 0.603·11-s + 0.832·13-s + 0.801·14-s − 1/4·16-s − 0.970·17-s − 0.917·19-s + 0.447·20-s + 0.426·22-s − 2.08·23-s − 8/5·25-s − 0.588·26-s + 0.566·28-s − 4.45·29-s − 0.718·31-s + 0.176·32-s + 0.685·34-s + 1.01·35-s + 0.648·38-s − 0.316·40-s − 0.312·41-s + 1.52·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{6} \cdot 7^{3} \cdot 13^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{6} \cdot 7^{3} \cdot 13^{3}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]
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 | 3 | | \( 1 \) |
| 7 | $C_1$ | \( ( 1 + T )^{3} \) |
| 13 | $C_1$ | \( ( 1 - T )^{3} \) |
| good | 2 | $S_4\times C_2$ | \( 1 + T + p T^{2} + p T^{3} + p^{2} T^{4} + p^{2} T^{5} + p^{3} T^{6} \) |
| 5 | $S_4\times C_2$ | \( 1 + 2 T + 12 T^{2} + 18 T^{3} + 12 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 11 | $S_4\times C_2$ | \( 1 + 2 T + 27 T^{2} + 36 T^{3} + 27 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 17 | $S_4\times C_2$ | \( 1 + 4 T + 41 T^{2} + 140 T^{3} + 41 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 19 | $S_4\times C_2$ | \( 1 + 4 T + 58 T^{2} + 148 T^{3} + 58 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 23 | $S_4\times C_2$ | \( 1 + 10 T + 70 T^{2} + 324 T^{3} + 70 p T^{4} + 10 p^{2} T^{5} + p^{3} T^{6} \) |
| 29 | $S_4\times C_2$ | \( 1 + 24 T + 272 T^{2} + 1846 T^{3} + 272 p T^{4} + 24 p^{2} T^{5} + p^{3} T^{6} \) |
| 31 | $S_4\times C_2$ | \( 1 + 4 T + 74 T^{2} + 264 T^{3} + 74 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 37 | $S_4\times C_2$ | \( 1 + 53 T^{2} - 124 T^{3} + 53 p T^{4} + p^{3} T^{6} \) |
| 41 | $S_4\times C_2$ | \( 1 + 2 T + 95 T^{2} + 172 T^{3} + 95 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 43 | $S_4\times C_2$ | \( 1 - 10 T + 58 T^{2} - 232 T^{3} + 58 p T^{4} - 10 p^{2} T^{5} + p^{3} T^{6} \) |
| 47 | $S_4\times C_2$ | \( 1 - 8 T + 62 T^{2} - 208 T^{3} + 62 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 53 | $S_4\times C_2$ | \( 1 + 8 T + 124 T^{2} + 870 T^{3} + 124 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 59 | $S_4\times C_2$ | \( 1 - 4 T + 21 T^{2} + 216 T^{3} + 21 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 61 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{3} \) |
| 67 | $S_4\times C_2$ | \( 1 + 12 T + 77 T^{2} + 632 T^{3} + 77 p T^{4} + 12 p^{2} T^{5} + p^{3} T^{6} \) |
| 71 | $S_4\times C_2$ | \( 1 - 6 T + 191 T^{2} - 868 T^{3} + 191 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 73 | $S_4\times C_2$ | \( 1 + 10 T + 120 T^{2} + 1186 T^{3} + 120 p T^{4} + 10 p^{2} T^{5} + p^{3} T^{6} \) |
| 79 | $S_4\times C_2$ | \( 1 + 14 T + 242 T^{2} + 2196 T^{3} + 242 p T^{4} + 14 p^{2} T^{5} + p^{3} T^{6} \) |
| 83 | $S_4\times C_2$ | \( 1 - 12 T - 22 T^{2} + 1276 T^{3} - 22 p T^{4} - 12 p^{2} T^{5} + p^{3} T^{6} \) |
| 89 | $S_4\times C_2$ | \( 1 + 2 T + 172 T^{2} - 66 T^{3} + 172 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 97 | $S_4\times C_2$ | \( 1 + 10 T + 320 T^{2} + 1962 T^{3} + 320 p T^{4} + 10 p^{2} T^{5} + p^{3} T^{6} \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{6} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.502522448235194953977555202779, −9.197734780899711612061834784288, −9.098150488521648811813385368875, −8.846617689363323918620099696931, −8.305181942635795144183105679228, −8.170684332921720965936983298486, −7.907917651429581848553319472586, −7.45885743492475338108392295467, −7.42109374514325990843542364952, −7.18409885839837209466190277603, −6.58321853954104567871249125807, −6.27237988664600452200797706210, −6.00029239396226175650572082314, −5.78564899618832235492388940253, −5.38737442781113332823979562466, −5.28907668433341721528533328361, −4.29346134336692432418724217176, −4.14660711147823997878544758739, −4.10834197149675344612278276159, −3.67573139932574585521627910032, −3.55768121705287165818097731263, −2.65660375596362281910892329588, −2.51146758915744126420069995012, −1.70917238687512570425534581137, −1.70117854442840435034473668166, 0, 0, 0,
1.70117854442840435034473668166, 1.70917238687512570425534581137, 2.51146758915744126420069995012, 2.65660375596362281910892329588, 3.55768121705287165818097731263, 3.67573139932574585521627910032, 4.10834197149675344612278276159, 4.14660711147823997878544758739, 4.29346134336692432418724217176, 5.28907668433341721528533328361, 5.38737442781113332823979562466, 5.78564899618832235492388940253, 6.00029239396226175650572082314, 6.27237988664600452200797706210, 6.58321853954104567871249125807, 7.18409885839837209466190277603, 7.42109374514325990843542364952, 7.45885743492475338108392295467, 7.907917651429581848553319472586, 8.170684332921720965936983298486, 8.305181942635795144183105679228, 8.846617689363323918620099696931, 9.098150488521648811813385368875, 9.197734780899711612061834784288, 9.502522448235194953977555202779
