| L(s) = 1 | + 2-s + 2.76·3-s + 4-s − 0.510·5-s + 2.76·6-s + 4.35·7-s + 8-s + 4.66·9-s − 0.510·10-s + 5.29·11-s + 2.76·12-s + 3.55·13-s + 4.35·14-s − 1.41·15-s + 16-s − 7.99·17-s + 4.66·18-s − 19-s − 0.510·20-s + 12.0·21-s + 5.29·22-s + 1.15·23-s + 2.76·24-s − 4.73·25-s + 3.55·26-s + 4.62·27-s + 4.35·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1.59·3-s + 0.5·4-s − 0.228·5-s + 1.13·6-s + 1.64·7-s + 0.353·8-s + 1.55·9-s − 0.161·10-s + 1.59·11-s + 0.799·12-s + 0.985·13-s + 1.16·14-s − 0.365·15-s + 0.250·16-s − 1.93·17-s + 1.10·18-s − 0.229·19-s − 0.114·20-s + 2.62·21-s + 1.12·22-s + 0.240·23-s + 0.565·24-s − 0.947·25-s + 0.696·26-s + 0.889·27-s + 0.822·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8018 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8018 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(7.962175010\) |
| \(L(\frac12)\) |
\(\approx\) |
\(7.962175010\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 19 | \( 1 + T \) |
| 211 | \( 1 + T \) |
| good | 3 | \( 1 - 2.76T + 3T^{2} \) |
| 5 | \( 1 + 0.510T + 5T^{2} \) |
| 7 | \( 1 - 4.35T + 7T^{2} \) |
| 11 | \( 1 - 5.29T + 11T^{2} \) |
| 13 | \( 1 - 3.55T + 13T^{2} \) |
| 17 | \( 1 + 7.99T + 17T^{2} \) |
| 23 | \( 1 - 1.15T + 23T^{2} \) |
| 29 | \( 1 + 1.94T + 29T^{2} \) |
| 31 | \( 1 + 8.99T + 31T^{2} \) |
| 37 | \( 1 - 8.54T + 37T^{2} \) |
| 41 | \( 1 - 3.50T + 41T^{2} \) |
| 43 | \( 1 - 9.98T + 43T^{2} \) |
| 47 | \( 1 + 5.88T + 47T^{2} \) |
| 53 | \( 1 - 10.9T + 53T^{2} \) |
| 59 | \( 1 + 11.0T + 59T^{2} \) |
| 61 | \( 1 + 8.35T + 61T^{2} \) |
| 67 | \( 1 + 3.77T + 67T^{2} \) |
| 71 | \( 1 + 11.2T + 71T^{2} \) |
| 73 | \( 1 - 7.77T + 73T^{2} \) |
| 79 | \( 1 + 1.43T + 79T^{2} \) |
| 83 | \( 1 + 3.11T + 83T^{2} \) |
| 89 | \( 1 - 0.187T + 89T^{2} \) |
| 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
−7.77527066017771033366909007431, −7.37149032122137220837971526619, −6.48471842839290923467630149789, −5.77420134812041039053890925195, −4.59075572655538049367289046645, −4.08306932143139241836312024250, −3.82828688461372327164262998655, −2.65165107033293633692101837211, −1.88894802987437777537699298985, −1.39355113516509952377521246430,
1.39355113516509952377521246430, 1.88894802987437777537699298985, 2.65165107033293633692101837211, 3.82828688461372327164262998655, 4.08306932143139241836312024250, 4.59075572655538049367289046645, 5.77420134812041039053890925195, 6.48471842839290923467630149789, 7.37149032122137220837971526619, 7.77527066017771033366909007431
