| L(s) = 1 | − 2.49·2-s − 1.70·3-s + 4.21·4-s − 1.58·5-s + 4.24·6-s + 2.99·7-s − 5.52·8-s − 0.103·9-s + 3.95·10-s + 1.39·11-s − 7.17·12-s − 6.59·13-s − 7.46·14-s + 2.69·15-s + 5.34·16-s + 4.23·17-s + 0.257·18-s − 3.71·19-s − 6.68·20-s − 5.09·21-s − 3.46·22-s − 5.72·23-s + 9.40·24-s − 2.48·25-s + 16.4·26-s + 5.28·27-s + 12.6·28-s + ⋯ |
| L(s) = 1 | − 1.76·2-s − 0.982·3-s + 2.10·4-s − 0.708·5-s + 1.73·6-s + 1.13·7-s − 1.95·8-s − 0.0343·9-s + 1.24·10-s + 0.419·11-s − 2.07·12-s − 1.82·13-s − 1.99·14-s + 0.696·15-s + 1.33·16-s + 1.02·17-s + 0.0606·18-s − 0.852·19-s − 1.49·20-s − 1.11·21-s − 0.739·22-s − 1.19·23-s + 1.91·24-s − 0.497·25-s + 3.22·26-s + 1.01·27-s + 2.38·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.1926542450\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1926542450\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 37 | \( 1 - T \) |
| 109 | \( 1 + T \) |
| good | 2 | \( 1 + 2.49T + 2T^{2} \) |
| 3 | \( 1 + 1.70T + 3T^{2} \) |
| 5 | \( 1 + 1.58T + 5T^{2} \) |
| 7 | \( 1 - 2.99T + 7T^{2} \) |
| 11 | \( 1 - 1.39T + 11T^{2} \) |
| 13 | \( 1 + 6.59T + 13T^{2} \) |
| 17 | \( 1 - 4.23T + 17T^{2} \) |
| 19 | \( 1 + 3.71T + 19T^{2} \) |
| 23 | \( 1 + 5.72T + 23T^{2} \) |
| 29 | \( 1 + 2.80T + 29T^{2} \) |
| 31 | \( 1 + 6.13T + 31T^{2} \) |
| 41 | \( 1 + 0.116T + 41T^{2} \) |
| 43 | \( 1 + 9.64T + 43T^{2} \) |
| 47 | \( 1 - 1.53T + 47T^{2} \) |
| 53 | \( 1 - 0.281T + 53T^{2} \) |
| 59 | \( 1 - 8.00T + 59T^{2} \) |
| 61 | \( 1 + 8.40T + 61T^{2} \) |
| 67 | \( 1 - 12.5T + 67T^{2} \) |
| 71 | \( 1 - 7.06T + 71T^{2} \) |
| 73 | \( 1 - 6.50T + 73T^{2} \) |
| 79 | \( 1 - 1.04T + 79T^{2} \) |
| 83 | \( 1 - 1.37T + 83T^{2} \) |
| 89 | \( 1 + 4.27T + 89T^{2} \) |
| 97 | \( 1 - 6.81T + 97T^{2} \) |
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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
−8.123668373300812650005403645890, −8.014856468134605742695748588852, −7.23452013975457581462200271527, −6.56101156359992513332349476763, −5.57199429455287059562657254249, −4.90232702265250615472084555350, −3.84080866085832997698137043739, −2.40450989243104567305643453634, −1.60997830607997662263565179938, −0.34335972115701459955411833586,
0.34335972115701459955411833586, 1.60997830607997662263565179938, 2.40450989243104567305643453634, 3.84080866085832997698137043739, 4.90232702265250615472084555350, 5.57199429455287059562657254249, 6.56101156359992513332349476763, 7.23452013975457581462200271527, 8.014856468134605742695748588852, 8.123668373300812650005403645890
