| L(s) = 1 | + 2-s + 4-s + 2.27·5-s + 3.85·7-s + 8-s + 2.27·10-s − 4.73·11-s + 3.85·13-s + 3.85·14-s + 16-s + 3.14·17-s + 4.89·19-s + 2.27·20-s − 4.73·22-s + 6.57·23-s + 0.166·25-s + 3.85·26-s + 3.85·28-s + 9.39·29-s − 0.492·31-s + 32-s + 3.14·34-s + 8.75·35-s − 4.90·37-s + 4.89·38-s + 2.27·40-s − 3.00·41-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.5·4-s + 1.01·5-s + 1.45·7-s + 0.353·8-s + 0.718·10-s − 1.42·11-s + 1.06·13-s + 1.02·14-s + 0.250·16-s + 0.763·17-s + 1.12·19-s + 0.508·20-s − 1.01·22-s + 1.36·23-s + 0.0332·25-s + 0.755·26-s + 0.727·28-s + 1.74·29-s − 0.0883·31-s + 0.176·32-s + 0.540·34-s + 1.47·35-s − 0.806·37-s + 0.794·38-s + 0.359·40-s − 0.469·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8046 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8046 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.437783075\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.437783075\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 3 | \( 1 \) |
| 149 | \( 1 + T \) |
| good | 5 | \( 1 - 2.27T + 5T^{2} \) |
| 7 | \( 1 - 3.85T + 7T^{2} \) |
| 11 | \( 1 + 4.73T + 11T^{2} \) |
| 13 | \( 1 - 3.85T + 13T^{2} \) |
| 17 | \( 1 - 3.14T + 17T^{2} \) |
| 19 | \( 1 - 4.89T + 19T^{2} \) |
| 23 | \( 1 - 6.57T + 23T^{2} \) |
| 29 | \( 1 - 9.39T + 29T^{2} \) |
| 31 | \( 1 + 0.492T + 31T^{2} \) |
| 37 | \( 1 + 4.90T + 37T^{2} \) |
| 41 | \( 1 + 3.00T + 41T^{2} \) |
| 43 | \( 1 + 4.16T + 43T^{2} \) |
| 47 | \( 1 - 1.86T + 47T^{2} \) |
| 53 | \( 1 + 1.94T + 53T^{2} \) |
| 59 | \( 1 + 9.17T + 59T^{2} \) |
| 61 | \( 1 + 15.1T + 61T^{2} \) |
| 67 | \( 1 + 14.1T + 67T^{2} \) |
| 71 | \( 1 - 11.0T + 71T^{2} \) |
| 73 | \( 1 + 14.7T + 73T^{2} \) |
| 79 | \( 1 + 13.8T + 79T^{2} \) |
| 83 | \( 1 - 1.47T + 83T^{2} \) |
| 89 | \( 1 - 14.8T + 89T^{2} \) |
| 97 | \( 1 + 11.4T + 97T^{2} \) |
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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
−7.72758396965548907875564131822, −7.20855080443704215875083108674, −6.13187475979023437943504958925, −5.67743089139709772358229972233, −4.89621569891124964289392910493, −4.75527709885126122214901220210, −3.30981576565276760848506265409, −2.82345135099852817611672075966, −1.73266670773984473526015579423, −1.18239525789331925847751010473,
1.18239525789331925847751010473, 1.73266670773984473526015579423, 2.82345135099852817611672075966, 3.30981576565276760848506265409, 4.75527709885126122214901220210, 4.89621569891124964289392910493, 5.67743089139709772358229972233, 6.13187475979023437943504958925, 7.20855080443704215875083108674, 7.72758396965548907875564131822
