| L(s) = 1 | + 2.43·2-s − 0.735·3-s + 3.90·4-s − 1.78·6-s + 0.581·7-s + 4.64·8-s − 2.45·9-s + 1.45·11-s − 2.87·12-s − 4.36·13-s + 1.41·14-s + 3.46·16-s − 5.97·18-s − 4.15·19-s − 0.427·21-s + 3.54·22-s + 7.01·23-s − 3.41·24-s − 10.6·26-s + 4.01·27-s + 2.27·28-s − 8.99·29-s − 7.73·31-s − 0.861·32-s − 1.07·33-s − 9.61·36-s + 1.58·37-s + ⋯ |
| L(s) = 1 | + 1.71·2-s − 0.424·3-s + 1.95·4-s − 0.729·6-s + 0.219·7-s + 1.64·8-s − 0.819·9-s + 0.439·11-s − 0.829·12-s − 1.21·13-s + 0.377·14-s + 0.866·16-s − 1.40·18-s − 0.953·19-s − 0.0933·21-s + 0.755·22-s + 1.46·23-s − 0.696·24-s − 2.08·26-s + 0.772·27-s + 0.429·28-s − 1.67·29-s − 1.38·31-s − 0.152·32-s − 0.186·33-s − 1.60·36-s + 0.260·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7225 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 17 | \( 1 \) |
| good | 2 | \( 1 - 2.43T + 2T^{2} \) |
| 3 | \( 1 + 0.735T + 3T^{2} \) |
| 7 | \( 1 - 0.581T + 7T^{2} \) |
| 11 | \( 1 - 1.45T + 11T^{2} \) |
| 13 | \( 1 + 4.36T + 13T^{2} \) |
| 19 | \( 1 + 4.15T + 19T^{2} \) |
| 23 | \( 1 - 7.01T + 23T^{2} \) |
| 29 | \( 1 + 8.99T + 29T^{2} \) |
| 31 | \( 1 + 7.73T + 31T^{2} \) |
| 37 | \( 1 - 1.58T + 37T^{2} \) |
| 41 | \( 1 - 4.05T + 41T^{2} \) |
| 43 | \( 1 - 4.54T + 43T^{2} \) |
| 47 | \( 1 + 1.40T + 47T^{2} \) |
| 53 | \( 1 + 10.6T + 53T^{2} \) |
| 59 | \( 1 - 0.999T + 59T^{2} \) |
| 61 | \( 1 + 5.87T + 61T^{2} \) |
| 67 | \( 1 + 1.24T + 67T^{2} \) |
| 71 | \( 1 - 13.8T + 71T^{2} \) |
| 73 | \( 1 + 9.01T + 73T^{2} \) |
| 79 | \( 1 - 7.50T + 79T^{2} \) |
| 83 | \( 1 + 15.8T + 83T^{2} \) |
| 89 | \( 1 + 7.36T + 89T^{2} \) |
| 97 | \( 1 + 13.0T + 97T^{2} \) |
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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
−7.21150837600782534631386199668, −6.67985198465356721201258505635, −5.89243611538071701298385958662, −5.38829387834134417559832377737, −4.79477396817136922395982298120, −4.12746483854034981030581457367, −3.26327967000530161858110366909, −2.57564896606667597282204347208, −1.71198842533060412523465096396, 0,
1.71198842533060412523465096396, 2.57564896606667597282204347208, 3.26327967000530161858110366909, 4.12746483854034981030581457367, 4.79477396817136922395982298120, 5.38829387834134417559832377737, 5.89243611538071701298385958662, 6.67985198465356721201258505635, 7.21150837600782534631386199668
