| L(s) = 1 | + 2·3-s − 4·7-s − 9-s + 2·11-s − 8·13-s − 4·17-s + 2·19-s − 8·21-s − 8·23-s − 4·27-s − 6·29-s − 8·31-s + 4·33-s − 8·37-s − 16·39-s + 2·41-s + 14·43-s − 4·47-s − 49-s − 8·51-s − 16·53-s + 4·57-s − 2·59-s − 10·61-s + 4·63-s + 10·67-s − 16·69-s + ⋯ |
| L(s) = 1 | + 1.15·3-s − 1.51·7-s − 1/3·9-s + 0.603·11-s − 2.21·13-s − 0.970·17-s + 0.458·19-s − 1.74·21-s − 1.66·23-s − 0.769·27-s − 1.11·29-s − 1.43·31-s + 0.696·33-s − 1.31·37-s − 2.56·39-s + 0.312·41-s + 2.13·43-s − 0.583·47-s − 1/7·49-s − 1.12·51-s − 2.19·53-s + 0.529·57-s − 0.260·59-s − 1.28·61-s + 0.503·63-s + 1.22·67-s − 1.92·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{21} \cdot 5^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{21} \cdot 5^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]
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 | | \( 1 \) |
| 5 | | \( 1 \) |
| good | 3 | $S_4\times C_2$ | \( 1 - 2 T + 5 T^{2} - 8 T^{3} + 5 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 7 | $S_4\times C_2$ | \( 1 + 4 T + 17 T^{2} + 36 T^{3} + 17 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 11 | $S_4\times C_2$ | \( 1 - 2 T + 13 T^{2} - 36 T^{3} + 13 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 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 | $S_4\times C_2$ | \( 1 + 4 T + 35 T^{2} + 104 T^{3} + 35 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 19 | $S_4\times C_2$ | \( 1 - 2 T + 21 T^{2} + 28 T^{3} + 21 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 23 | $S_4\times C_2$ | \( 1 + 8 T + 81 T^{2} + 372 T^{3} + 81 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 29 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{3} \) |
| 31 | $S_4\times C_2$ | \( 1 + 8 T + 61 T^{2} + 368 T^{3} + 61 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 37 | $S_4\times C_2$ | \( 1 + 8 T + 79 T^{2} + 320 T^{3} + 79 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 41 | $S_4\times C_2$ | \( 1 - 2 T + 71 T^{2} + 20 T^{3} + 71 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 43 | $S_4\times C_2$ | \( 1 - 14 T + 149 T^{2} - 1104 T^{3} + 149 p T^{4} - 14 p^{2} T^{5} + p^{3} T^{6} \) |
| 47 | $S_4\times C_2$ | \( 1 + 4 T + 113 T^{2} + 260 T^{3} + 113 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 53 | $S_4\times C_2$ | \( 1 + 16 T + 231 T^{2} + 1776 T^{3} + 231 p T^{4} + 16 p^{2} T^{5} + p^{3} T^{6} \) |
| 59 | $S_4\times C_2$ | \( 1 + 2 T + 141 T^{2} + 132 T^{3} + 141 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 61 | $S_4\times C_2$ | \( 1 + 10 T + 155 T^{2} + 1212 T^{3} + 155 p T^{4} + 10 p^{2} T^{5} + p^{3} T^{6} \) |
| 67 | $S_4\times C_2$ | \( 1 - 10 T + 141 T^{2} - 736 T^{3} + 141 p T^{4} - 10 p^{2} T^{5} + p^{3} T^{6} \) |
| 71 | $S_4\times C_2$ | \( 1 - 12 T + 197 T^{2} - 1384 T^{3} + 197 p T^{4} - 12 p^{2} T^{5} + p^{3} T^{6} \) |
| 73 | $S_4\times C_2$ | \( 1 + 20 T + 3 p T^{2} + 1736 T^{3} + 3 p^{2} T^{4} + 20 p^{2} T^{5} + p^{3} T^{6} \) |
| 79 | $S_4\times C_2$ | \( 1 + 16 T + 269 T^{2} + 2400 T^{3} + 269 p T^{4} + 16 p^{2} T^{5} + p^{3} T^{6} \) |
| 83 | $S_4\times C_2$ | \( 1 - 30 T + 509 T^{2} - 5504 T^{3} + 509 p T^{4} - 30 p^{2} T^{5} + p^{3} T^{6} \) |
| 89 | $S_4\times C_2$ | \( 1 + 10 T + 151 T^{2} + 684 T^{3} + 151 p T^{4} + 10 p^{2} T^{5} + p^{3} T^{6} \) |
| 97 | $S_4\times C_2$ | \( 1 + 28 T + 531 T^{2} + 6040 T^{3} + 531 p T^{4} + 28 p^{2} T^{5} + p^{3} T^{6} \) |
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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
−8.231500550476992298001379586884, −7.83447485222623908373309285918, −7.49254558651499496415527980314, −7.29719028093854592966903450231, −7.17690945470824178476222254495, −6.80653949907176271771846352974, −6.68299208929099535511810146973, −6.16134791289940731557917992864, −6.08587757693125828610539297517, −5.97236527366127025351165533014, −5.32234051715402591864990752976, −5.29000042424312757704363048280, −5.09571223162272905484156378113, −4.53456042811516392832498522765, −4.24380414513320110897871360516, −4.14821748877366509900620166900, −3.62612730076007288214873434246, −3.52893386632641623073122104097, −3.19645066753924958876180117968, −2.78807006481003564942604121219, −2.66936833990450223717936013331, −2.39192631925079358840252824701, −1.91037322653510538411315666572, −1.76117582241263531733954057404, −1.19537397572838742088773266482, 0, 0, 0,
1.19537397572838742088773266482, 1.76117582241263531733954057404, 1.91037322653510538411315666572, 2.39192631925079358840252824701, 2.66936833990450223717936013331, 2.78807006481003564942604121219, 3.19645066753924958876180117968, 3.52893386632641623073122104097, 3.62612730076007288214873434246, 4.14821748877366509900620166900, 4.24380414513320110897871360516, 4.53456042811516392832498522765, 5.09571223162272905484156378113, 5.29000042424312757704363048280, 5.32234051715402591864990752976, 5.97236527366127025351165533014, 6.08587757693125828610539297517, 6.16134791289940731557917992864, 6.68299208929099535511810146973, 6.80653949907176271771846352974, 7.17690945470824178476222254495, 7.29719028093854592966903450231, 7.49254558651499496415527980314, 7.83447485222623908373309285918, 8.231500550476992298001379586884
