| L(s) = 1 | − 2-s + 3.06·3-s + 4-s + 5-s − 3.06·6-s − 3.10·7-s − 8-s + 6.39·9-s − 10-s − 3.44·11-s + 3.06·12-s − 13-s + 3.10·14-s + 3.06·15-s + 16-s − 3.16·17-s − 6.39·18-s − 1.99·19-s + 20-s − 9.52·21-s + 3.44·22-s − 4.04·23-s − 3.06·24-s + 25-s + 26-s + 10.4·27-s − 3.10·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1.77·3-s + 0.5·4-s + 0.447·5-s − 1.25·6-s − 1.17·7-s − 0.353·8-s + 2.13·9-s − 0.316·10-s − 1.03·11-s + 0.885·12-s − 0.277·13-s + 0.830·14-s + 0.791·15-s + 0.250·16-s − 0.767·17-s − 1.50·18-s − 0.457·19-s + 0.223·20-s − 2.07·21-s + 0.734·22-s − 0.843·23-s − 0.625·24-s + 0.200·25-s + 0.196·26-s + 2.00·27-s − 0.586·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4030 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4030 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 5 | \( 1 - T \) |
| 13 | \( 1 + T \) |
| 31 | \( 1 - T \) |
| good | 3 | \( 1 - 3.06T + 3T^{2} \) |
| 7 | \( 1 + 3.10T + 7T^{2} \) |
| 11 | \( 1 + 3.44T + 11T^{2} \) |
| 17 | \( 1 + 3.16T + 17T^{2} \) |
| 19 | \( 1 + 1.99T + 19T^{2} \) |
| 23 | \( 1 + 4.04T + 23T^{2} \) |
| 29 | \( 1 + 3.65T + 29T^{2} \) |
| 37 | \( 1 + 0.111T + 37T^{2} \) |
| 41 | \( 1 + 7.09T + 41T^{2} \) |
| 43 | \( 1 + 4.15T + 43T^{2} \) |
| 47 | \( 1 + 4.18T + 47T^{2} \) |
| 53 | \( 1 + 0.478T + 53T^{2} \) |
| 59 | \( 1 + 6.41T + 59T^{2} \) |
| 61 | \( 1 - 1.74T + 61T^{2} \) |
| 67 | \( 1 - 9.25T + 67T^{2} \) |
| 71 | \( 1 - 11.6T + 71T^{2} \) |
| 73 | \( 1 + 10.6T + 73T^{2} \) |
| 79 | \( 1 - 7.38T + 79T^{2} \) |
| 83 | \( 1 + 16.8T + 83T^{2} \) |
| 89 | \( 1 - 3.02T + 89T^{2} \) |
| 97 | \( 1 - 2.17T + 97T^{2} \) |
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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
−8.236471231644671544484555805644, −7.57628794621929083407017257519, −6.83684205536993252053318728198, −6.19818267972574487476624850277, −5.00784891012983687914433324230, −3.87634062965321948895742114787, −3.12241424780903533989535869519, −2.43942704268107530307775751953, −1.79882300342338140032190811886, 0,
1.79882300342338140032190811886, 2.43942704268107530307775751953, 3.12241424780903533989535869519, 3.87634062965321948895742114787, 5.00784891012983687914433324230, 6.19818267972574487476624850277, 6.83684205536993252053318728198, 7.57628794621929083407017257519, 8.236471231644671544484555805644
