| L(s) = 1 | − 2-s − 2.38·3-s + 4-s + 1.49·5-s + 2.38·6-s − 1.41·7-s − 8-s + 2.67·9-s − 1.49·10-s − 1.07·11-s − 2.38·12-s + 0.488·13-s + 1.41·14-s − 3.55·15-s + 16-s − 7.39·17-s − 2.67·18-s + 3.62·19-s + 1.49·20-s + 3.37·21-s + 1.07·22-s − 2.34·23-s + 2.38·24-s − 2.77·25-s − 0.488·26-s + 0.778·27-s − 1.41·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.37·3-s + 0.5·4-s + 0.666·5-s + 0.972·6-s − 0.535·7-s − 0.353·8-s + 0.891·9-s − 0.471·10-s − 0.323·11-s − 0.687·12-s + 0.135·13-s + 0.378·14-s − 0.916·15-s + 0.250·16-s − 1.79·17-s − 0.630·18-s + 0.831·19-s + 0.333·20-s + 0.735·21-s + 0.228·22-s − 0.488·23-s + 0.486·24-s − 0.555·25-s − 0.0957·26-s + 0.149·27-s − 0.267·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4034 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4034 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4889104837\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4889104837\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 2017 | \( 1 - T \) |
| good | 3 | \( 1 + 2.38T + 3T^{2} \) |
| 5 | \( 1 - 1.49T + 5T^{2} \) |
| 7 | \( 1 + 1.41T + 7T^{2} \) |
| 11 | \( 1 + 1.07T + 11T^{2} \) |
| 13 | \( 1 - 0.488T + 13T^{2} \) |
| 17 | \( 1 + 7.39T + 17T^{2} \) |
| 19 | \( 1 - 3.62T + 19T^{2} \) |
| 23 | \( 1 + 2.34T + 23T^{2} \) |
| 29 | \( 1 - 8.34T + 29T^{2} \) |
| 31 | \( 1 + 8.27T + 31T^{2} \) |
| 37 | \( 1 + 9.34T + 37T^{2} \) |
| 41 | \( 1 - 0.0389T + 41T^{2} \) |
| 43 | \( 1 - 9.90T + 43T^{2} \) |
| 47 | \( 1 - 5.58T + 47T^{2} \) |
| 53 | \( 1 + 13.0T + 53T^{2} \) |
| 59 | \( 1 - 6.50T + 59T^{2} \) |
| 61 | \( 1 - 3.60T + 61T^{2} \) |
| 67 | \( 1 - 11.4T + 67T^{2} \) |
| 71 | \( 1 + 6.86T + 71T^{2} \) |
| 73 | \( 1 - 6.61T + 73T^{2} \) |
| 79 | \( 1 - 12.0T + 79T^{2} \) |
| 83 | \( 1 + 6.00T + 83T^{2} \) |
| 89 | \( 1 + 4.43T + 89T^{2} \) |
| 97 | \( 1 + 8.20T + 97T^{2} \) |
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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
−8.589162535712481430764761865446, −7.59349705450528959146250026846, −6.70724661869828425511312826508, −6.39924203395733797756932936415, −5.59384919785294201785887869757, −5.00908469234674720835492348568, −3.91844658613227012666981454974, −2.67818421797397912917254917486, −1.73274403813251991190609278904, −0.46784392256185536877282116861,
0.46784392256185536877282116861, 1.73274403813251991190609278904, 2.67818421797397912917254917486, 3.91844658613227012666981454974, 5.00908469234674720835492348568, 5.59384919785294201785887869757, 6.39924203395733797756932936415, 6.70724661869828425511312826508, 7.59349705450528959146250026846, 8.589162535712481430764761865446
