| L(s) = 1 | + 9.78·2-s − 9·3-s + 63.7·4-s + 84.2·5-s − 88.0·6-s − 182.·7-s + 311.·8-s + 81·9-s + 824.·10-s − 152.·11-s − 574.·12-s − 1.10e3·13-s − 1.78e3·14-s − 757.·15-s + 1.00e3·16-s − 1.62e3·17-s + 792.·18-s + 472.·19-s + 5.37e3·20-s + 1.64e3·21-s − 1.49e3·22-s − 3.96e3·23-s − 2.79e3·24-s + 3.96e3·25-s − 1.07e4·26-s − 729·27-s − 1.16e4·28-s + ⋯ |
| L(s) = 1 | + 1.73·2-s − 0.577·3-s + 1.99·4-s + 1.50·5-s − 0.998·6-s − 1.40·7-s + 1.71·8-s + 0.333·9-s + 2.60·10-s − 0.380·11-s − 1.15·12-s − 1.81·13-s − 2.43·14-s − 0.869·15-s + 0.979·16-s − 1.36·17-s + 0.576·18-s + 0.300·19-s + 3.00·20-s + 0.812·21-s − 0.657·22-s − 1.56·23-s − 0.992·24-s + 1.26·25-s − 3.13·26-s − 0.192·27-s − 2.80·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 471 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 471 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + 9T \) |
| 157 | \( 1 + 2.46e4T \) |
| good | 2 | \( 1 - 9.78T + 32T^{2} \) |
| 5 | \( 1 - 84.2T + 3.12e3T^{2} \) |
| 7 | \( 1 + 182.T + 1.68e4T^{2} \) |
| 11 | \( 1 + 152.T + 1.61e5T^{2} \) |
| 13 | \( 1 + 1.10e3T + 3.71e5T^{2} \) |
| 17 | \( 1 + 1.62e3T + 1.41e6T^{2} \) |
| 19 | \( 1 - 472.T + 2.47e6T^{2} \) |
| 23 | \( 1 + 3.96e3T + 6.43e6T^{2} \) |
| 29 | \( 1 - 3.57e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 4.26e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 5.75e3T + 6.93e7T^{2} \) |
| 41 | \( 1 + 1.99e4T + 1.15e8T^{2} \) |
| 43 | \( 1 - 1.71e4T + 1.47e8T^{2} \) |
| 47 | \( 1 - 2.61e4T + 2.29e8T^{2} \) |
| 53 | \( 1 - 2.59e4T + 4.18e8T^{2} \) |
| 59 | \( 1 + 4.76e3T + 7.14e8T^{2} \) |
| 61 | \( 1 + 4.41e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 3.38e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 5.83e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 5.54e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 5.07e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 2.54e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 1.01e5T + 5.58e9T^{2} \) |
| 97 | \( 1 + 1.94e4T + 8.58e9T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.00441591621735744448232368479, −9.285454889006401073140452094827, −7.23538101762968054414963469364, −6.51444375022014465156845053279, −5.85953099241007046259994329459, −5.15018769477524536380630700528, −4.13830843005557897392325918484, −2.69309634598702925651142797455, −2.15680350996619613099872754162, 0,
2.15680350996619613099872754162, 2.69309634598702925651142797455, 4.13830843005557897392325918484, 5.15018769477524536380630700528, 5.85953099241007046259994329459, 6.51444375022014465156845053279, 7.23538101762968054414963469364, 9.285454889006401073140452094827, 10.00441591621735744448232368479
