| L(s) = 1 | − 2-s − 4-s − 4·5-s + 4·10-s + 9·13-s + 16-s + 2·17-s − 9·19-s + 4·20-s − 11·23-s + 2·25-s − 9·26-s + 6·29-s + 3·31-s + 2·32-s − 2·34-s − 3·37-s + 9·38-s − 13·41-s − 6·43-s + 11·46-s + 47-s − 12·49-s − 2·50-s − 9·52-s − 15·53-s − 6·58-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1/2·4-s − 1.78·5-s + 1.26·10-s + 2.49·13-s + 1/4·16-s + 0.485·17-s − 2.06·19-s + 0.894·20-s − 2.29·23-s + 2/5·25-s − 1.76·26-s + 1.11·29-s + 0.538·31-s + 0.353·32-s − 0.342·34-s − 0.493·37-s + 1.45·38-s − 2.03·41-s − 0.914·43-s + 1.62·46-s + 0.145·47-s − 1.71·49-s − 0.282·50-s − 1.24·52-s − 2.06·53-s − 0.787·58-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{9} \cdot 11^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{9} \cdot 11^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]
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 | | \( 1 \) |
| 11 | | \( 1 \) |
| good | 2 | $S_4\times C_2$ | \( 1 + T + p T^{2} + 3 T^{3} + p^{2} T^{4} + p^{2} T^{5} + p^{3} T^{6} \) |
| 5 | $S_4\times C_2$ | \( 1 + 4 T + 14 T^{2} + 39 T^{3} + 14 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 7 | $S_4\times C_2$ | \( 1 + 12 T^{2} - T^{3} + 12 p T^{4} + p^{3} T^{6} \) |
| 13 | $S_4\times C_2$ | \( 1 - 9 T + 3 p T^{2} - 127 T^{3} + 3 p^{2} T^{4} - 9 p^{2} T^{5} + p^{3} T^{6} \) |
| 17 | $S_4\times C_2$ | \( 1 - 2 T + 26 T^{2} - 9 T^{3} + 26 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 19 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{3} \) |
| 23 | $S_4\times C_2$ | \( 1 + 11 T + 83 T^{2} + 417 T^{3} + 83 p T^{4} + 11 p^{2} T^{5} + p^{3} T^{6} \) |
| 29 | $S_4\times C_2$ | \( 1 - 6 T + 63 T^{2} - 276 T^{3} + 63 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 31 | $S_4\times C_2$ | \( 1 - 3 T + 27 T^{2} + 71 T^{3} + 27 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 37 | $S_4\times C_2$ | \( 1 + 3 T + 87 T^{2} + 249 T^{3} + 87 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 41 | $S_4\times C_2$ | \( 1 + 13 T + 125 T^{2} + 1023 T^{3} + 125 p T^{4} + 13 p^{2} T^{5} + p^{3} T^{6} \) |
| 43 | $S_4\times C_2$ | \( 1 + 6 T + 114 T^{2} + 417 T^{3} + 114 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 47 | $S_4\times C_2$ | \( 1 - T + 92 T^{2} + 27 T^{3} + 92 p T^{4} - p^{2} T^{5} + p^{3} T^{6} \) |
| 53 | $S_4\times C_2$ | \( 1 + 15 T + 207 T^{2} + 1527 T^{3} + 207 p T^{4} + 15 p^{2} T^{5} + p^{3} T^{6} \) |
| 59 | $S_4\times C_2$ | \( 1 + 17 T + 167 T^{2} + 1269 T^{3} + 167 p T^{4} + 17 p^{2} T^{5} + p^{3} T^{6} \) |
| 61 | $S_4\times C_2$ | \( 1 + 15 T + 249 T^{2} + 1911 T^{3} + 249 p T^{4} + 15 p^{2} T^{5} + p^{3} T^{6} \) |
| 67 | $S_4\times C_2$ | \( 1 - 15 T + 207 T^{2} - 1601 T^{3} + 207 p T^{4} - 15 p^{2} T^{5} + p^{3} T^{6} \) |
| 71 | $S_4\times C_2$ | \( 1 + 21 T + 351 T^{2} + 3261 T^{3} + 351 p T^{4} + 21 p^{2} T^{5} + p^{3} T^{6} \) |
| 73 | $S_4\times C_2$ | \( 1 + 12 T + 210 T^{2} + 1451 T^{3} + 210 p T^{4} + 12 p^{2} T^{5} + p^{3} T^{6} \) |
| 79 | $S_4\times C_2$ | \( 1 + 9 T + 93 T^{2} + 523 T^{3} + 93 p T^{4} + 9 p^{2} T^{5} + p^{3} T^{6} \) |
| 83 | $S_4\times C_2$ | \( 1 - 15 T + 141 T^{2} - 789 T^{3} + 141 p T^{4} - 15 p^{2} T^{5} + p^{3} T^{6} \) |
| 89 | $S_4\times C_2$ | \( 1 - 3 T + 261 T^{2} - 525 T^{3} + 261 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 97 | $S_4\times C_2$ | \( 1 + 9 T + 162 T^{2} + 1393 T^{3} + 162 p T^{4} + 9 p^{2} T^{5} + p^{3} T^{6} \) |
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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
−8.204375577741539805905391496348, −7.918796292283634318282975590120, −7.79218965140110176240108104138, −7.42449227994341860832457707603, −7.16436943282177614203006203197, −6.67310861662956754718582629558, −6.35876439770523109009450693317, −6.31436370108180109139326098809, −6.15367300845571991592322561305, −6.01929781817812614984408401040, −5.57769631650298160676301303105, −5.03726217621722105135733057415, −4.87555150265810344007912857675, −4.51248355399816640552656244334, −4.25311546728134242695827803486, −4.21982522672568026674234858940, −3.66433204197327163612176072218, −3.62049628291257130159528517282, −3.36071661853778916950165734583, −2.94684077108203844122089774163, −2.79830402632950067577343065381, −1.83882784875641966019294920200, −1.78361125904791392605556007052, −1.54945105301474026487199101039, −0.949205263139479937073782570064, 0, 0, 0,
0.949205263139479937073782570064, 1.54945105301474026487199101039, 1.78361125904791392605556007052, 1.83882784875641966019294920200, 2.79830402632950067577343065381, 2.94684077108203844122089774163, 3.36071661853778916950165734583, 3.62049628291257130159528517282, 3.66433204197327163612176072218, 4.21982522672568026674234858940, 4.25311546728134242695827803486, 4.51248355399816640552656244334, 4.87555150265810344007912857675, 5.03726217621722105135733057415, 5.57769631650298160676301303105, 6.01929781817812614984408401040, 6.15367300845571991592322561305, 6.31436370108180109139326098809, 6.35876439770523109009450693317, 6.67310861662956754718582629558, 7.16436943282177614203006203197, 7.42449227994341860832457707603, 7.79218965140110176240108104138, 7.918796292283634318282975590120, 8.204375577741539805905391496348
