| L(s) = 1 | − 3·3-s + 2·5-s − 3·7-s + 9-s − 7·11-s + 2·13-s − 6·15-s + 7·17-s − 3·19-s + 9·21-s + 14·23-s − 6·25-s + 7·27-s + 3·29-s − 11·31-s + 21·33-s − 6·35-s − 6·39-s − 7·41-s + 4·43-s + 2·45-s + 8·47-s + 6·49-s − 21·51-s + 53-s − 14·55-s + 9·57-s + ⋯ |
| L(s) = 1 | − 1.73·3-s + 0.894·5-s − 1.13·7-s + 1/3·9-s − 2.11·11-s + 0.554·13-s − 1.54·15-s + 1.69·17-s − 0.688·19-s + 1.96·21-s + 2.91·23-s − 6/5·25-s + 1.34·27-s + 0.557·29-s − 1.97·31-s + 3.65·33-s − 1.01·35-s − 0.960·39-s − 1.09·41-s + 0.609·43-s + 0.298·45-s + 1.16·47-s + 6/7·49-s − 2.94·51-s + 0.137·53-s − 1.88·55-s + 1.19·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{18} \cdot 7^{3} \cdot 19^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{18} \cdot 7^{3} \cdot 19^{3}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]
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 | | \( 1 \) |
| 7 | $C_1$ | \( ( 1 + T )^{3} \) |
| 19 | $C_1$ | \( ( 1 + T )^{3} \) |
| good | 3 | $S_4\times C_2$ | \( 1 + p T + 8 T^{2} + 14 T^{3} + 8 p T^{4} + p^{3} T^{5} + p^{3} T^{6} \) |
| 5 | $S_4\times C_2$ | \( 1 - 2 T + 2 p T^{2} - 18 T^{3} + 2 p^{2} T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 11 | $S_4\times C_2$ | \( 1 + 7 T + 4 p T^{2} + 158 T^{3} + 4 p^{2} T^{4} + 7 p^{2} T^{5} + p^{3} T^{6} \) |
| 13 | $S_4\times C_2$ | \( 1 - 2 T + 34 T^{2} - 50 T^{3} + 34 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 17 | $S_4\times C_2$ | \( 1 - 7 T + 40 T^{2} - 132 T^{3} + 40 p T^{4} - 7 p^{2} T^{5} + p^{3} T^{6} \) |
| 23 | $S_4\times C_2$ | \( 1 - 14 T + 122 T^{2} - 700 T^{3} + 122 p T^{4} - 14 p^{2} T^{5} + p^{3} T^{6} \) |
| 29 | $S_4\times C_2$ | \( 1 - 3 T + 14 T^{2} + 104 T^{3} + 14 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 31 | $S_4\times C_2$ | \( 1 + 11 T + 118 T^{2} + 698 T^{3} + 118 p T^{4} + 11 p^{2} T^{5} + p^{3} T^{6} \) |
| 37 | $S_4\times C_2$ | \( 1 + 68 T^{2} - 106 T^{3} + 68 p T^{4} + p^{3} T^{6} \) |
| 41 | $S_4\times C_2$ | \( 1 + 7 T - 28 T^{2} - 424 T^{3} - 28 p T^{4} + 7 p^{2} T^{5} + p^{3} T^{6} \) |
| 43 | $S_4\times C_2$ | \( 1 - 4 T + 109 T^{2} - 328 T^{3} + 109 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 47 | $S_4\times C_2$ | \( 1 - 8 T + 112 T^{2} - 768 T^{3} + 112 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 53 | $S_4\times C_2$ | \( 1 - T + 128 T^{2} - 104 T^{3} + 128 p T^{4} - p^{2} T^{5} + p^{3} T^{6} \) |
| 59 | $S_4\times C_2$ | \( 1 - 10 T + 178 T^{2} - 1056 T^{3} + 178 p T^{4} - 10 p^{2} T^{5} + p^{3} T^{6} \) |
| 61 | $S_4\times C_2$ | \( 1 - 6 T + 134 T^{2} - 650 T^{3} + 134 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 67 | $S_4\times C_2$ | \( 1 - 3 T + 122 T^{2} - 214 T^{3} + 122 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 71 | $S_4\times C_2$ | \( 1 + 152 T^{2} - 32 T^{3} + 152 p T^{4} + p^{3} T^{6} \) |
| 73 | $S_4\times C_2$ | \( 1 - T + 118 T^{2} - 244 T^{3} + 118 p T^{4} - p^{2} T^{5} + p^{3} T^{6} \) |
| 79 | $S_4\times C_2$ | \( 1 + 4 T + 193 T^{2} + 664 T^{3} + 193 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 83 | $S_4\times C_2$ | \( 1 + 31 T + 538 T^{2} + 5934 T^{3} + 538 p T^{4} + 31 p^{2} T^{5} + p^{3} T^{6} \) |
| 89 | $S_4\times C_2$ | \( 1 + 28 T + 371 T^{2} + 3632 T^{3} + 371 p T^{4} + 28 p^{2} T^{5} + p^{3} T^{6} \) |
| 97 | $S_4\times C_2$ | \( 1 + 30 T + 534 T^{2} + 6302 T^{3} + 534 p T^{4} + 30 p^{2} T^{5} + p^{3} T^{6} \) |
| 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
−7.04784960504765531724880436671, −6.94213463281379938370616646878, −6.74015556110371023771053295938, −6.64351077901225897167722689450, −6.00618233763961226824705477669, −5.92812597983482153953793207022, −5.76791300880834702054866508529, −5.55690467379824512708195171073, −5.55564935763584390836687770564, −5.22371767619098008728360456176, −5.14100914060403736165816493811, −4.94436152225991249063541907881, −4.46571306500220432770236896418, −3.96348447361102782335170800345, −3.87729262164020350839383637659, −3.84074380917077995482881589127, −3.11297827182873428832739330777, −3.05691732576515462258334008097, −2.75063440134278432635368865278, −2.60550629150678625556847380484, −2.41380858491889938727993007836, −1.84872416717301476528816980793, −1.31803266948190901420974015913, −1.24742382205146565032243506579, −0.863009802359048103698523724741, 0, 0, 0,
0.863009802359048103698523724741, 1.24742382205146565032243506579, 1.31803266948190901420974015913, 1.84872416717301476528816980793, 2.41380858491889938727993007836, 2.60550629150678625556847380484, 2.75063440134278432635368865278, 3.05691732576515462258334008097, 3.11297827182873428832739330777, 3.84074380917077995482881589127, 3.87729262164020350839383637659, 3.96348447361102782335170800345, 4.46571306500220432770236896418, 4.94436152225991249063541907881, 5.14100914060403736165816493811, 5.22371767619098008728360456176, 5.55564935763584390836687770564, 5.55690467379824512708195171073, 5.76791300880834702054866508529, 5.92812597983482153953793207022, 6.00618233763961226824705477669, 6.64351077901225897167722689450, 6.74015556110371023771053295938, 6.94213463281379938370616646878, 7.04784960504765531724880436671
