| L(s) = 1 | − 2-s + 1.32·3-s + 4-s + 2.16·5-s − 1.32·6-s − 2.98·7-s − 8-s − 1.25·9-s − 2.16·10-s + 2.12·11-s + 1.32·12-s + 2.25·13-s + 2.98·14-s + 2.85·15-s + 16-s + 1.94·17-s + 1.25·18-s − 4.50·19-s + 2.16·20-s − 3.94·21-s − 2.12·22-s − 2.25·23-s − 1.32·24-s − 0.312·25-s − 2.25·26-s − 5.62·27-s − 2.98·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.762·3-s + 0.5·4-s + 0.968·5-s − 0.538·6-s − 1.13·7-s − 0.353·8-s − 0.419·9-s − 0.684·10-s + 0.641·11-s + 0.381·12-s + 0.626·13-s + 0.799·14-s + 0.737·15-s + 0.250·16-s + 0.472·17-s + 0.296·18-s − 1.03·19-s + 0.484·20-s − 0.861·21-s − 0.453·22-s − 0.469·23-s − 0.269·24-s − 0.0625·25-s − 0.442·26-s − 1.08·27-s − 0.565·28-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
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 2003 | \( 1 + T \) |
| good | 3 | \( 1 - 1.32T + 3T^{2} \) |
| 5 | \( 1 - 2.16T + 5T^{2} \) |
| 7 | \( 1 + 2.98T + 7T^{2} \) |
| 11 | \( 1 - 2.12T + 11T^{2} \) |
| 13 | \( 1 - 2.25T + 13T^{2} \) |
| 17 | \( 1 - 1.94T + 17T^{2} \) |
| 19 | \( 1 + 4.50T + 19T^{2} \) |
| 23 | \( 1 + 2.25T + 23T^{2} \) |
| 29 | \( 1 + 3.38T + 29T^{2} \) |
| 31 | \( 1 + 6.92T + 31T^{2} \) |
| 37 | \( 1 + 2.51T + 37T^{2} \) |
| 41 | \( 1 + 2.47T + 41T^{2} \) |
| 43 | \( 1 + 5.35T + 43T^{2} \) |
| 47 | \( 1 - 2.67T + 47T^{2} \) |
| 53 | \( 1 + 7.74T + 53T^{2} \) |
| 59 | \( 1 - 8.45T + 59T^{2} \) |
| 61 | \( 1 + 7.65T + 61T^{2} \) |
| 67 | \( 1 - 0.113T + 67T^{2} \) |
| 71 | \( 1 - 5.80T + 71T^{2} \) |
| 73 | \( 1 + 3.14T + 73T^{2} \) |
| 79 | \( 1 - 13.9T + 79T^{2} \) |
| 83 | \( 1 - 8.08T + 83T^{2} \) |
| 89 | \( 1 + 9.85T + 89T^{2} \) |
| 97 | \( 1 + 15.6T + 97T^{2} \) |
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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
−8.332941854280332126832166159773, −7.45977077372082843160813503027, −6.51912227742671325536221966451, −6.13242863064396602505494373265, −5.38375551722953541186562764052, −3.83365974726459686771584511414, −3.32664459622169101835432620091, −2.30849907494635608727497267921, −1.61172384905982804628450271105, 0,
1.61172384905982804628450271105, 2.30849907494635608727497267921, 3.32664459622169101835432620091, 3.83365974726459686771584511414, 5.38375551722953541186562764052, 6.13242863064396602505494373265, 6.51912227742671325536221966451, 7.45977077372082843160813503027, 8.332941854280332126832166159773
