| L(s) = 1 | + 0.901·2-s + 0.927·3-s − 31.1·4-s − 25·5-s + 0.836·6-s − 27.0·7-s − 56.9·8-s − 242.·9-s − 22.5·10-s − 121·11-s − 28.9·12-s + 833.·13-s − 24.3·14-s − 23.1·15-s + 946.·16-s − 2.15e3·17-s − 218.·18-s − 361·19-s + 779.·20-s − 25.0·21-s − 109.·22-s + 4.04e3·23-s − 52.8·24-s + 625·25-s + 750.·26-s − 450.·27-s + 843.·28-s + ⋯ |
| L(s) = 1 | + 0.159·2-s + 0.0595·3-s − 0.974·4-s − 0.447·5-s + 0.00948·6-s − 0.208·7-s − 0.314·8-s − 0.996·9-s − 0.0712·10-s − 0.301·11-s − 0.0580·12-s + 1.36·13-s − 0.0332·14-s − 0.0266·15-s + 0.924·16-s − 1.81·17-s − 0.158·18-s − 0.229·19-s + 0.435·20-s − 0.0124·21-s − 0.0480·22-s + 1.59·23-s − 0.0187·24-s + 0.200·25-s + 0.217·26-s − 0.118·27-s + 0.203·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 + 25T \) |
| 11 | \( 1 + 121T \) |
| 19 | \( 1 + 361T \) |
| good | 2 | \( 1 - 0.901T + 32T^{2} \) |
| 3 | \( 1 - 0.927T + 243T^{2} \) |
| 7 | \( 1 + 27.0T + 1.68e4T^{2} \) |
| 13 | \( 1 - 833.T + 3.71e5T^{2} \) |
| 17 | \( 1 + 2.15e3T + 1.41e6T^{2} \) |
| 23 | \( 1 - 4.04e3T + 6.43e6T^{2} \) |
| 29 | \( 1 - 2.65e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 378.T + 2.86e7T^{2} \) |
| 37 | \( 1 + 9.94e3T + 6.93e7T^{2} \) |
| 41 | \( 1 - 5.75e3T + 1.15e8T^{2} \) |
| 43 | \( 1 - 7.54e3T + 1.47e8T^{2} \) |
| 47 | \( 1 - 2.37e4T + 2.29e8T^{2} \) |
| 53 | \( 1 - 2.30e4T + 4.18e8T^{2} \) |
| 59 | \( 1 - 9.84e3T + 7.14e8T^{2} \) |
| 61 | \( 1 - 1.00e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 2.72e3T + 1.35e9T^{2} \) |
| 71 | \( 1 - 1.81e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 6.04e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 2.19e3T + 3.07e9T^{2} \) |
| 83 | \( 1 - 9.49e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 8.86e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 3.44e4T + 8.58e9T^{2} \) |
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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
−8.737042024000086573820811746136, −8.322649180459822228616415092046, −7.04667364747150949456816044414, −6.13289148025078230524575120847, −5.24932846156744610416485646459, −4.34456126621830394656740635598, −3.52338155986208337637624988872, −2.57318660513683950798908271962, −0.928134917913993783131331817013, 0,
0.928134917913993783131331817013, 2.57318660513683950798908271962, 3.52338155986208337637624988872, 4.34456126621830394656740635598, 5.24932846156744610416485646459, 6.13289148025078230524575120847, 7.04667364747150949456816044414, 8.322649180459822228616415092046, 8.737042024000086573820811746136
