| L(s) = 1 | − 2-s + 1.90·3-s + 4-s + 4.10·5-s − 1.90·6-s − 0.345·7-s − 8-s + 0.640·9-s − 4.10·10-s − 5.57·11-s + 1.90·12-s + 2.72·13-s + 0.345·14-s + 7.83·15-s + 16-s + 4.30·17-s − 0.640·18-s + 4.51·19-s + 4.10·20-s − 0.659·21-s + 5.57·22-s + 1.20·23-s − 1.90·24-s + 11.8·25-s − 2.72·26-s − 4.50·27-s − 0.345·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1.10·3-s + 0.5·4-s + 1.83·5-s − 0.778·6-s − 0.130·7-s − 0.353·8-s + 0.213·9-s − 1.29·10-s − 1.68·11-s + 0.550·12-s + 0.754·13-s + 0.0923·14-s + 2.02·15-s + 0.250·16-s + 1.04·17-s − 0.150·18-s + 1.03·19-s + 0.918·20-s − 0.143·21-s + 1.18·22-s + 0.252·23-s − 0.389·24-s + 2.37·25-s − 0.533·26-s − 0.866·27-s − 0.0652·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4006 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4006 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.900936473\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.900936473\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 2003 | \( 1 - T \) |
| good | 3 | \( 1 - 1.90T + 3T^{2} \) |
| 5 | \( 1 - 4.10T + 5T^{2} \) |
| 7 | \( 1 + 0.345T + 7T^{2} \) |
| 11 | \( 1 + 5.57T + 11T^{2} \) |
| 13 | \( 1 - 2.72T + 13T^{2} \) |
| 17 | \( 1 - 4.30T + 17T^{2} \) |
| 19 | \( 1 - 4.51T + 19T^{2} \) |
| 23 | \( 1 - 1.20T + 23T^{2} \) |
| 29 | \( 1 - 2.88T + 29T^{2} \) |
| 31 | \( 1 + 1.75T + 31T^{2} \) |
| 37 | \( 1 - 7.25T + 37T^{2} \) |
| 41 | \( 1 - 1.17T + 41T^{2} \) |
| 43 | \( 1 + 0.951T + 43T^{2} \) |
| 47 | \( 1 - 9.22T + 47T^{2} \) |
| 53 | \( 1 - 5.28T + 53T^{2} \) |
| 59 | \( 1 - 2.62T + 59T^{2} \) |
| 61 | \( 1 + 12.8T + 61T^{2} \) |
| 67 | \( 1 + 11.1T + 67T^{2} \) |
| 71 | \( 1 + 16.2T + 71T^{2} \) |
| 73 | \( 1 - 1.74T + 73T^{2} \) |
| 79 | \( 1 - 2.65T + 79T^{2} \) |
| 83 | \( 1 - 9.19T + 83T^{2} \) |
| 89 | \( 1 - 7.38T + 89T^{2} \) |
| 97 | \( 1 - 10.4T + 97T^{2} \) |
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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
−8.601569712467899736011207136054, −7.79379453593978810972842395820, −7.33341977045955095589922127949, −6.04182683911225426144109538487, −5.77070283059962738526436046644, −4.89211182114608376776487627189, −3.24774764613903554405586279726, −2.80825407722192288984933697111, −2.06288725918776808458405416994, −1.08275486990172096763270185038,
1.08275486990172096763270185038, 2.06288725918776808458405416994, 2.80825407722192288984933697111, 3.24774764613903554405586279726, 4.89211182114608376776487627189, 5.77070283059962738526436046644, 6.04182683911225426144109538487, 7.33341977045955095589922127949, 7.79379453593978810972842395820, 8.601569712467899736011207136054
