| L(s) = 1 | − 3-s − 3·5-s − 7-s − 4·9-s − 11·13-s + 3·15-s − 3·17-s + 3·19-s + 21-s − 9·23-s + 6·25-s + 5·27-s − 7·29-s + 6·31-s + 3·35-s − 20·37-s + 11·39-s − 22·41-s − 10·43-s + 12·45-s − 4·49-s + 3·51-s − 7·53-s − 3·57-s + 11·59-s − 16·61-s + 4·63-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 1.34·5-s − 0.377·7-s − 4/3·9-s − 3.05·13-s + 0.774·15-s − 0.727·17-s + 0.688·19-s + 0.218·21-s − 1.87·23-s + 6/5·25-s + 0.962·27-s − 1.29·29-s + 1.07·31-s + 0.507·35-s − 3.28·37-s + 1.76·39-s − 3.43·41-s − 1.52·43-s + 1.78·45-s − 4/7·49-s + 0.420·51-s − 0.961·53-s − 0.397·57-s + 1.43·59-s − 2.04·61-s + 0.503·63-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{9} \cdot 5^{3} \cdot 19^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{9} \cdot 5^{3} \cdot 19^{3}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]
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 | | \( 1 \) |
| 5 | $C_1$ | \( ( 1 + T )^{3} \) |
| 19 | $C_1$ | \( ( 1 - T )^{3} \) |
| good | 3 | $S_4\times C_2$ | \( 1 + T + 5 T^{2} + 4 T^{3} + 5 p T^{4} + p^{2} T^{5} + p^{3} T^{6} \) |
| 7 | $S_4\times C_2$ | \( 1 + T + 5 T^{2} + 30 T^{3} + 5 p T^{4} + p^{2} T^{5} + p^{3} T^{6} \) |
| 11 | $S_4\times C_2$ | \( 1 + 5 T^{2} + 16 T^{3} + 5 p T^{4} + p^{3} T^{6} \) |
| 13 | $S_4\times C_2$ | \( 1 + 11 T + 55 T^{2} + 200 T^{3} + 55 p T^{4} + 11 p^{2} T^{5} + p^{3} T^{6} \) |
| 17 | $S_4\times C_2$ | \( 1 + 3 T + 47 T^{2} + 98 T^{3} + 47 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 23 | $S_4\times C_2$ | \( 1 + 9 T + 89 T^{2} + 422 T^{3} + 89 p T^{4} + 9 p^{2} T^{5} + p^{3} T^{6} \) |
| 29 | $S_4\times C_2$ | \( 1 + 7 T + 43 T^{2} + 114 T^{3} + 43 p T^{4} + 7 p^{2} T^{5} + p^{3} T^{6} \) |
| 31 | $S_4\times C_2$ | \( 1 - 6 T + 77 T^{2} - 340 T^{3} + 77 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 37 | $S_4\times C_2$ | \( 1 + 20 T + 237 T^{2} + 1724 T^{3} + 237 p T^{4} + 20 p^{2} T^{5} + p^{3} T^{6} \) |
| 41 | $D_{6}$ | \( 1 + 22 T + 223 T^{2} + 1572 T^{3} + 223 p T^{4} + 22 p^{2} T^{5} + p^{3} T^{6} \) |
| 43 | $S_4\times C_2$ | \( 1 + 10 T + 97 T^{2} + 508 T^{3} + 97 p T^{4} + 10 p^{2} T^{5} + p^{3} T^{6} \) |
| 47 | $S_4\times C_2$ | \( 1 + 29 T^{2} - 128 T^{3} + 29 p T^{4} + p^{3} T^{6} \) |
| 53 | $S_4\times C_2$ | \( 1 + 7 T - 9 T^{2} - 600 T^{3} - 9 p T^{4} + 7 p^{2} T^{5} + p^{3} T^{6} \) |
| 59 | $S_4\times C_2$ | \( 1 - 11 T + 37 T^{2} + 246 T^{3} + 37 p T^{4} - 11 p^{2} T^{5} + p^{3} T^{6} \) |
| 61 | $S_4\times C_2$ | \( 1 + 16 T + 207 T^{2} + 1600 T^{3} + 207 p T^{4} + 16 p^{2} T^{5} + p^{3} T^{6} \) |
| 67 | $S_4\times C_2$ | \( 1 + T + 101 T^{2} + 396 T^{3} + 101 p T^{4} + p^{2} T^{5} + p^{3} T^{6} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{3} \) |
| 73 | $S_4\times C_2$ | \( 1 + 5 T + 47 T^{2} - 498 T^{3} + 47 p T^{4} + 5 p^{2} T^{5} + p^{3} T^{6} \) |
| 79 | $S_4\times C_2$ | \( 1 - 26 T + 445 T^{2} - 4604 T^{3} + 445 p T^{4} - 26 p^{2} T^{5} + p^{3} T^{6} \) |
| 83 | $S_4\times C_2$ | \( 1 + 14 T + 297 T^{2} + 2340 T^{3} + 297 p T^{4} + 14 p^{2} T^{5} + p^{3} T^{6} \) |
| 89 | $S_4\times C_2$ | \( 1 + 6 T + 15 T^{2} - 188 T^{3} + 15 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 97 | $S_4\times C_2$ | \( 1 - 8 T + 233 T^{2} - 1260 T^{3} + 233 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} \) |
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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
−9.851047708190152447872137914912, −9.300111935762805548800554094466, −9.082066331624621691716459984824, −8.737575313337994366301470568337, −8.320629452269819123312144686836, −8.228861095782982207773762068028, −8.012238407729892217142472895756, −7.53096182092576360959363346625, −7.24289138361354677131164445580, −7.11895386741679432208888755894, −6.69824582957521889411372400506, −6.42565122341162467266792565573, −6.26440952286459499269251298369, −5.42176104799751120212455301215, −5.31852128526798038013231331601, −5.27819174808657037261681585591, −4.71825770333991849318671689267, −4.62080971355182172365566202781, −4.07072846440327463695774985273, −3.51360915669260130052113253599, −3.25415039825249590894571527107, −3.15140318969988800211125564551, −2.46246116481331798869274685032, −2.02150641146546994728047556882, −1.66190957001146624107476081496, 0, 0, 0,
1.66190957001146624107476081496, 2.02150641146546994728047556882, 2.46246116481331798869274685032, 3.15140318969988800211125564551, 3.25415039825249590894571527107, 3.51360915669260130052113253599, 4.07072846440327463695774985273, 4.62080971355182172365566202781, 4.71825770333991849318671689267, 5.27819174808657037261681585591, 5.31852128526798038013231331601, 5.42176104799751120212455301215, 6.26440952286459499269251298369, 6.42565122341162467266792565573, 6.69824582957521889411372400506, 7.11895386741679432208888755894, 7.24289138361354677131164445580, 7.53096182092576360959363346625, 8.012238407729892217142472895756, 8.228861095782982207773762068028, 8.320629452269819123312144686836, 8.737575313337994366301470568337, 9.082066331624621691716459984824, 9.300111935762805548800554094466, 9.851047708190152447872137914912
