| L(s) = 1 | + 3·2-s − 4·3-s + 6·4-s + 5-s − 12·6-s − 4·7-s + 10·8-s + 7·9-s + 3·10-s + 3·11-s − 24·12-s − 4·13-s − 12·14-s − 4·15-s + 15·16-s + 5·17-s + 21·18-s + 6·20-s + 16·21-s + 9·22-s − 9·23-s − 40·24-s − 7·25-s − 12·26-s − 9·27-s − 24·28-s − 6·29-s + ⋯ |
| L(s) = 1 | + 2.12·2-s − 2.30·3-s + 3·4-s + 0.447·5-s − 4.89·6-s − 1.51·7-s + 3.53·8-s + 7/3·9-s + 0.948·10-s + 0.904·11-s − 6.92·12-s − 1.10·13-s − 3.20·14-s − 1.03·15-s + 15/4·16-s + 1.21·17-s + 4.94·18-s + 1.34·20-s + 3.49·21-s + 1.91·22-s − 1.87·23-s − 8.16·24-s − 7/5·25-s − 2.35·26-s − 1.73·27-s − 4.53·28-s − 1.11·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{3} \cdot 11^{3} \cdot 19^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{3} \cdot 11^{3} \cdot 19^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]
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 | 2 | $C_1$ | \( ( 1 - T )^{3} \) |
| 11 | $C_1$ | \( ( 1 - T )^{3} \) |
| 19 | | \( 1 \) |
| good | 3 | $S_4\times C_2$ | \( 1 + 4 T + p^{2} T^{2} + 17 T^{3} + p^{3} T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 5 | $S_4\times C_2$ | \( 1 - T + 8 T^{2} - 14 T^{3} + 8 p T^{4} - p^{2} T^{5} + p^{3} T^{6} \) |
| 7 | $S_4\times C_2$ | \( 1 + 4 T + 19 T^{2} + 55 T^{3} + 19 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 13 | $S_4\times C_2$ | \( 1 + 4 T + 37 T^{2} + 103 T^{3} + 37 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 17 | $S_4\times C_2$ | \( 1 - 5 T + 47 T^{2} - 154 T^{3} + 47 p T^{4} - 5 p^{2} T^{5} + p^{3} T^{6} \) |
| 23 | $S_4\times C_2$ | \( 1 + 9 T + 83 T^{2} + 386 T^{3} + 83 p T^{4} + 9 p^{2} T^{5} + p^{3} T^{6} \) |
| 29 | $S_4\times C_2$ | \( 1 + 6 T + 51 T^{2} + 199 T^{3} + 51 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 31 | $S_4\times C_2$ | \( 1 - T + 58 T^{2} - 34 T^{3} + 58 p T^{4} - p^{2} T^{5} + p^{3} T^{6} \) |
| 37 | $S_4\times C_2$ | \( 1 - 8 T + 103 T^{2} - 584 T^{3} + 103 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 41 | $S_4\times C_2$ | \( 1 + 19 T + 238 T^{2} + 1776 T^{3} + 238 p T^{4} + 19 p^{2} T^{5} + p^{3} T^{6} \) |
| 43 | $S_4\times C_2$ | \( 1 - 17 T + 218 T^{2} - 1596 T^{3} + 218 p T^{4} - 17 p^{2} T^{5} + p^{3} T^{6} \) |
| 47 | $S_4\times C_2$ | \( 1 - 2 T + 121 T^{2} - 156 T^{3} + 121 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 53 | $S_4\times C_2$ | \( 1 - 3 T + 45 T^{2} - 634 T^{3} + 45 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 59 | $S_4\times C_2$ | \( 1 - 11 T + 161 T^{2} - 1234 T^{3} + 161 p T^{4} - 11 p^{2} T^{5} + p^{3} T^{6} \) |
| 61 | $S_4\times C_2$ | \( 1 + 71 T^{2} + 392 T^{3} + 71 p T^{4} + p^{3} T^{6} \) |
| 67 | $S_4\times C_2$ | \( 1 - 2 T + T^{2} + 13 p T^{3} + p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 71 | $S_4\times C_2$ | \( 1 + 21 T + 332 T^{2} + 3178 T^{3} + 332 p T^{4} + 21 p^{2} T^{5} + p^{3} T^{6} \) |
| 73 | $S_4\times C_2$ | \( 1 + 3 T + 105 T^{2} + 754 T^{3} + 105 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 79 | $S_4\times C_2$ | \( 1 + 101 T^{2} + 488 T^{3} + 101 p T^{4} + p^{3} T^{6} \) |
| 83 | $S_4\times C_2$ | \( 1 + 25 T + 428 T^{2} + 4538 T^{3} + 428 p T^{4} + 25 p^{2} T^{5} + p^{3} T^{6} \) |
| 89 | $S_4\times C_2$ | \( 1 + 4 T + 155 T^{2} + 128 T^{3} + 155 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 97 | $S_4\times C_2$ | \( 1 - 10 T + 275 T^{2} - 1812 T^{3} + 275 p T^{4} - 10 p^{2} T^{5} + p^{3} T^{6} \) |
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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
−7.06779158540551608907845759423, −6.76702466781359308665893029752, −6.63711218481427899498257483513, −6.45494819639210878329758548536, −6.16055721244281346070926013074, −5.99083029160062676595568121839, −5.74626206316864676111294829022, −5.63444113682637198953039599590, −5.57786147567076735026536503699, −5.37609876932743869280986353484, −4.94841493234602860586942052236, −4.79387990642661571717190003726, −4.49651187635058709103408961572, −3.98979555575227800826730275993, −3.98087724336153254298910590350, −3.89428840849438615241807297746, −3.66845426652614838021528793216, −3.14326883361856332678416226102, −2.94969288711924715899503035372, −2.50024271971199573254973490971, −2.47730415408409314802530359475, −2.03176843446715995498853686319, −1.58118300557524949156001164956, −1.28812541683869522396092959757, −1.10363842198347840992297333642, 0, 0, 0,
1.10363842198347840992297333642, 1.28812541683869522396092959757, 1.58118300557524949156001164956, 2.03176843446715995498853686319, 2.47730415408409314802530359475, 2.50024271971199573254973490971, 2.94969288711924715899503035372, 3.14326883361856332678416226102, 3.66845426652614838021528793216, 3.89428840849438615241807297746, 3.98087724336153254298910590350, 3.98979555575227800826730275993, 4.49651187635058709103408961572, 4.79387990642661571717190003726, 4.94841493234602860586942052236, 5.37609876932743869280986353484, 5.57786147567076735026536503699, 5.63444113682637198953039599590, 5.74626206316864676111294829022, 5.99083029160062676595568121839, 6.16055721244281346070926013074, 6.45494819639210878329758548536, 6.63711218481427899498257483513, 6.76702466781359308665893029752, 7.06779158540551608907845759423
