| L(s) = 1 | − 3·2-s + 6·4-s − 2·5-s − 3·7-s − 10·8-s + 6·10-s − 6·11-s + 8·13-s + 9·14-s + 15·16-s − 3·19-s − 12·20-s + 18·22-s − 4·23-s + 25-s − 24·26-s − 18·28-s − 2·29-s − 2·31-s − 21·32-s + 6·35-s + 4·37-s + 9·38-s + 20·40-s − 2·41-s − 4·43-s − 36·44-s + ⋯ |
| L(s) = 1 | − 2.12·2-s + 3·4-s − 0.894·5-s − 1.13·7-s − 3.53·8-s + 1.89·10-s − 1.80·11-s + 2.21·13-s + 2.40·14-s + 15/4·16-s − 0.688·19-s − 2.68·20-s + 3.83·22-s − 0.834·23-s + 1/5·25-s − 4.70·26-s − 3.40·28-s − 0.371·29-s − 0.359·31-s − 3.71·32-s + 1.01·35-s + 0.657·37-s + 1.45·38-s + 3.16·40-s − 0.312·41-s − 0.609·43-s − 5.42·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{3} \cdot 3^{6} \cdot 7^{3} \cdot 19^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{3} \cdot 3^{6} \cdot 7^{3} \cdot 19^{3}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]
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 | $C_1$ | \( ( 1 + T )^{3} \) |
| 3 | | \( 1 \) |
| 7 | $C_1$ | \( ( 1 + T )^{3} \) |
| 19 | $C_1$ | \( ( 1 + T )^{3} \) |
| good | 5 | $D_{6}$ | \( 1 + 2 T + 3 T^{2} + 12 T^{3} + 3 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 11 | $S_4\times C_2$ | \( 1 + 6 T + 29 T^{2} + 92 T^{3} + 29 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 13 | $S_4\times C_2$ | \( 1 - 8 T + 47 T^{2} - 192 T^{3} + 47 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 17 | $S_4\times C_2$ | \( 1 + 35 T^{2} + 16 T^{3} + 35 p T^{4} + p^{3} T^{6} \) |
| 23 | $S_4\times C_2$ | \( 1 + 4 T + 61 T^{2} + 168 T^{3} + 61 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 29 | $D_{6}$ | \( 1 + 2 T + 3 T^{2} + 12 T^{3} + 3 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 31 | $S_4\times C_2$ | \( 1 + 2 T + 73 T^{2} + 116 T^{3} + 73 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 37 | $S_4\times C_2$ | \( 1 - 4 T + 95 T^{2} - 264 T^{3} + 95 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 41 | $S_4\times C_2$ | \( 1 + 2 T + 71 T^{2} + 124 T^{3} + 71 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 43 | $S_4\times C_2$ | \( 1 + 4 T + 49 T^{2} + 280 T^{3} + 49 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 47 | $S_4\times C_2$ | \( 1 + 16 T + 189 T^{2} + 1536 T^{3} + 189 p T^{4} + 16 p^{2} T^{5} + p^{3} T^{6} \) |
| 53 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{3} \) |
| 59 | $C_2$ | \( ( 1 + p T^{2} )^{3} \) |
| 61 | $S_4\times C_2$ | \( 1 - 10 T + 131 T^{2} - 1020 T^{3} + 131 p T^{4} - 10 p^{2} T^{5} + p^{3} T^{6} \) |
| 67 | $S_4\times C_2$ | \( 1 + 185 T^{2} + 16 T^{3} + 185 p T^{4} + p^{3} T^{6} \) |
| 71 | $S_4\times C_2$ | \( 1 + 24 T + 341 T^{2} + 3280 T^{3} + 341 p T^{4} + 24 p^{2} T^{5} + p^{3} T^{6} \) |
| 73 | $S_4\times C_2$ | \( 1 + 6 T + 167 T^{2} + 628 T^{3} + 167 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 79 | $S_4\times C_2$ | \( 1 + 18 T + 113 T^{2} + 308 T^{3} + 113 p T^{4} + 18 p^{2} T^{5} + p^{3} T^{6} \) |
| 83 | $S_4\times C_2$ | \( 1 + 14 T + 261 T^{2} + 2172 T^{3} + 261 p T^{4} + 14 p^{2} T^{5} + p^{3} T^{6} \) |
| 89 | $S_4\times C_2$ | \( 1 + 18 T + 311 T^{2} + 3164 T^{3} + 311 p T^{4} + 18 p^{2} T^{5} + p^{3} T^{6} \) |
| 97 | $S_4\times C_2$ | \( 1 + 2 T + 207 T^{2} + 620 T^{3} + 207 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) |
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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
−8.326791736313744317283855217183, −8.135151797854797868173012544632, −7.930758925643898739930161589968, −7.88782969938713066852762723371, −7.15785253902148509270748822171, −7.13653806034022389690660277513, −7.13405090346856709925366450982, −6.60026205672845120958643387714, −6.31566218450742317583931689535, −6.17728924446077865220249220285, −5.76809128296442863145846140806, −5.64037384329443643719702928499, −5.47533088235218677825775953295, −4.85288394234750914436769599629, −4.38308827613506632925333422912, −4.33429237159046649161289146098, −3.71616194795567500093413605449, −3.53096621715463464290854091966, −3.31866939533966687567480939472, −2.90431723380468291136910743686, −2.52169912523752188001534879569, −2.46161432495221159675463209423, −1.64608593932271902264090282161, −1.44093003567167674418071237242, −1.15191837687881531423443140446, 0, 0, 0,
1.15191837687881531423443140446, 1.44093003567167674418071237242, 1.64608593932271902264090282161, 2.46161432495221159675463209423, 2.52169912523752188001534879569, 2.90431723380468291136910743686, 3.31866939533966687567480939472, 3.53096621715463464290854091966, 3.71616194795567500093413605449, 4.33429237159046649161289146098, 4.38308827613506632925333422912, 4.85288394234750914436769599629, 5.47533088235218677825775953295, 5.64037384329443643719702928499, 5.76809128296442863145846140806, 6.17728924446077865220249220285, 6.31566218450742317583931689535, 6.60026205672845120958643387714, 7.13405090346856709925366450982, 7.13653806034022389690660277513, 7.15785253902148509270748822171, 7.88782969938713066852762723371, 7.930758925643898739930161589968, 8.135151797854797868173012544632, 8.326791736313744317283855217183
