| L(s) = 1 | − 2.44·2-s − 0.0495·3-s + 3.99·4-s + 1.48·5-s + 0.121·6-s + 2.04·7-s − 4.89·8-s − 2.99·9-s − 3.64·10-s + 4.87·11-s − 0.198·12-s − 13-s − 5.01·14-s − 0.0737·15-s + 3.99·16-s − 3.11·17-s + 7.34·18-s + 3.69·19-s + 5.95·20-s − 0.101·21-s − 11.9·22-s + 5.02·23-s + 0.242·24-s − 2.78·25-s + 2.44·26-s + 0.297·27-s + 8.19·28-s + ⋯ |
| L(s) = 1 | − 1.73·2-s − 0.0285·3-s + 1.99·4-s + 0.665·5-s + 0.0495·6-s + 0.774·7-s − 1.73·8-s − 0.999·9-s − 1.15·10-s + 1.46·11-s − 0.0571·12-s − 0.277·13-s − 1.34·14-s − 0.0190·15-s + 0.998·16-s − 0.754·17-s + 1.73·18-s + 0.846·19-s + 1.33·20-s − 0.0221·21-s − 2.54·22-s + 1.04·23-s + 0.0495·24-s − 0.556·25-s + 0.480·26-s + 0.0571·27-s + 1.54·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1339 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1339 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9165444085\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9165444085\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 13 | \( 1 + T \) |
| 103 | \( 1 - T \) |
| good | 2 | \( 1 + 2.44T + 2T^{2} \) |
| 3 | \( 1 + 0.0495T + 3T^{2} \) |
| 5 | \( 1 - 1.48T + 5T^{2} \) |
| 7 | \( 1 - 2.04T + 7T^{2} \) |
| 11 | \( 1 - 4.87T + 11T^{2} \) |
| 17 | \( 1 + 3.11T + 17T^{2} \) |
| 19 | \( 1 - 3.69T + 19T^{2} \) |
| 23 | \( 1 - 5.02T + 23T^{2} \) |
| 29 | \( 1 - 8.90T + 29T^{2} \) |
| 31 | \( 1 - 4.75T + 31T^{2} \) |
| 37 | \( 1 + 9.83T + 37T^{2} \) |
| 41 | \( 1 + 4.13T + 41T^{2} \) |
| 43 | \( 1 - 8.39T + 43T^{2} \) |
| 47 | \( 1 - 5.37T + 47T^{2} \) |
| 53 | \( 1 - 1.21T + 53T^{2} \) |
| 59 | \( 1 - 12.5T + 59T^{2} \) |
| 61 | \( 1 + 1.83T + 61T^{2} \) |
| 67 | \( 1 + 13.6T + 67T^{2} \) |
| 71 | \( 1 + 6.56T + 71T^{2} \) |
| 73 | \( 1 + 7.13T + 73T^{2} \) |
| 79 | \( 1 - 5.31T + 79T^{2} \) |
| 83 | \( 1 + 3.44T + 83T^{2} \) |
| 89 | \( 1 - 0.953T + 89T^{2} \) |
| 97 | \( 1 - 14.6T + 97T^{2} \) |
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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
−9.422864988664623860107093767540, −8.801184637802739939899055360118, −8.404244968696933123198192016126, −7.29172458719871915190403073609, −6.63511974034375899960591004621, −5.77495372925712083509752192574, −4.61515097440988076431608523254, −2.99124124919934689433901371364, −1.93369737539673916546859229924, −0.949343570001377499855955101484,
0.949343570001377499855955101484, 1.93369737539673916546859229924, 2.99124124919934689433901371364, 4.61515097440988076431608523254, 5.77495372925712083509752192574, 6.63511974034375899960591004621, 7.29172458719871915190403073609, 8.404244968696933123198192016126, 8.801184637802739939899055360118, 9.422864988664623860107093767540
