| L(s) = 1 | − 2-s + 1.38·3-s + 4-s − 5-s − 1.38·6-s + 4.92·7-s − 8-s − 1.09·9-s + 10-s − 0.449·11-s + 1.38·12-s − 6.45·13-s − 4.92·14-s − 1.38·15-s + 16-s − 0.359·17-s + 1.09·18-s − 1.64·19-s − 20-s + 6.79·21-s + 0.449·22-s − 0.251·23-s − 1.38·24-s + 25-s + 6.45·26-s − 5.65·27-s + 4.92·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.797·3-s + 0.5·4-s − 0.447·5-s − 0.563·6-s + 1.85·7-s − 0.353·8-s − 0.364·9-s + 0.316·10-s − 0.135·11-s + 0.398·12-s − 1.79·13-s − 1.31·14-s − 0.356·15-s + 0.250·16-s − 0.0872·17-s + 0.257·18-s − 0.376·19-s − 0.223·20-s + 1.48·21-s + 0.0957·22-s − 0.0525·23-s − 0.281·24-s + 0.200·25-s + 1.26·26-s − 1.08·27-s + 0.929·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6010 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6010 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 601 | \( 1 + T \) |
| good | 3 | \( 1 - 1.38T + 3T^{2} \) |
| 7 | \( 1 - 4.92T + 7T^{2} \) |
| 11 | \( 1 + 0.449T + 11T^{2} \) |
| 13 | \( 1 + 6.45T + 13T^{2} \) |
| 17 | \( 1 + 0.359T + 17T^{2} \) |
| 19 | \( 1 + 1.64T + 19T^{2} \) |
| 23 | \( 1 + 0.251T + 23T^{2} \) |
| 29 | \( 1 - 6.77T + 29T^{2} \) |
| 31 | \( 1 + 1.11T + 31T^{2} \) |
| 37 | \( 1 + 1.81T + 37T^{2} \) |
| 41 | \( 1 + 2.12T + 41T^{2} \) |
| 43 | \( 1 + 3.24T + 43T^{2} \) |
| 47 | \( 1 - 3.00T + 47T^{2} \) |
| 53 | \( 1 + 12.2T + 53T^{2} \) |
| 59 | \( 1 - 6.99T + 59T^{2} \) |
| 61 | \( 1 + 10.5T + 61T^{2} \) |
| 67 | \( 1 - 10.7T + 67T^{2} \) |
| 71 | \( 1 + 3.62T + 71T^{2} \) |
| 73 | \( 1 - 6.33T + 73T^{2} \) |
| 79 | \( 1 + 5.26T + 79T^{2} \) |
| 83 | \( 1 + 12.5T + 83T^{2} \) |
| 89 | \( 1 + 5.95T + 89T^{2} \) |
| 97 | \( 1 + 1.15T + 97T^{2} \) |
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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
−7.87943784178170104507027095918, −7.43300680523073640228465489423, −6.61057096729489676579604070814, −5.37421521133002047435268565346, −4.86500705608697816450686443119, −4.07111595115813738971197078216, −2.86258183302781719404532792838, −2.30956858380563694550994131947, −1.44946163129015612923369314640, 0,
1.44946163129015612923369314640, 2.30956858380563694550994131947, 2.86258183302781719404532792838, 4.07111595115813738971197078216, 4.86500705608697816450686443119, 5.37421521133002047435268565346, 6.61057096729489676579604070814, 7.43300680523073640228465489423, 7.87943784178170104507027095918
