L(s) = 1 | − 2-s − 1.91·3-s + 4-s − 1.39·5-s + 1.91·6-s + 3.46·7-s − 8-s + 0.652·9-s + 1.39·10-s − 5.38·11-s − 1.91·12-s + 0.707·13-s − 3.46·14-s + 2.67·15-s + 16-s + 1.30·17-s − 0.652·18-s + 1.25·19-s − 1.39·20-s − 6.62·21-s + 5.38·22-s − 1.63·23-s + 1.91·24-s − 3.04·25-s − 0.707·26-s + 4.48·27-s + 3.46·28-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 1.10·3-s + 0.5·4-s − 0.624·5-s + 0.780·6-s + 1.31·7-s − 0.353·8-s + 0.217·9-s + 0.441·10-s − 1.62·11-s − 0.551·12-s + 0.196·13-s − 0.927·14-s + 0.689·15-s + 0.250·16-s + 0.315·17-s − 0.153·18-s + 0.287·19-s − 0.312·20-s − 1.44·21-s + 1.14·22-s − 0.340·23-s + 0.390·24-s − 0.609·25-s − 0.138·26-s + 0.863·27-s + 0.655·28-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 | \( 1 + 1.91T + 3T^{2} \) |
| 5 | \( 1 + 1.39T + 5T^{2} \) |
| 7 | \( 1 - 3.46T + 7T^{2} \) |
| 11 | \( 1 + 5.38T + 11T^{2} \) |
| 13 | \( 1 - 0.707T + 13T^{2} \) |
| 17 | \( 1 - 1.30T + 17T^{2} \) |
| 19 | \( 1 - 1.25T + 19T^{2} \) |
| 23 | \( 1 + 1.63T + 23T^{2} \) |
| 29 | \( 1 - 9.27T + 29T^{2} \) |
| 31 | \( 1 - 3.30T + 31T^{2} \) |
| 37 | \( 1 + 2.58T + 37T^{2} \) |
| 41 | \( 1 + 5.07T + 41T^{2} \) |
| 43 | \( 1 - 4.23T + 43T^{2} \) |
| 47 | \( 1 + 9.82T + 47T^{2} \) |
| 53 | \( 1 - 2.68T + 53T^{2} \) |
| 59 | \( 1 + 4.53T + 59T^{2} \) |
| 61 | \( 1 + 9.89T + 61T^{2} \) |
| 67 | \( 1 + 7.12T + 67T^{2} \) |
| 71 | \( 1 - 15.6T + 71T^{2} \) |
| 73 | \( 1 - 7.20T + 73T^{2} \) |
| 79 | \( 1 - 12.9T + 79T^{2} \) |
| 83 | \( 1 - 1.68T + 83T^{2} \) |
| 89 | \( 1 - 14.6T + 89T^{2} \) |
| 97 | \( 1 + 4.62T + 97T^{2} \) |
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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
−8.129146925387390203492825972053, −7.60461143813055641389754872174, −6.64444742424468432147445062151, −5.84936444694027856373721378685, −5.04122383286582937255022309137, −4.67740386295752647696242363002, −3.29923494252027306647894563727, −2.26045931126908092584692228526, −1.06811128757607116262477863287, 0,
1.06811128757607116262477863287, 2.26045931126908092584692228526, 3.29923494252027306647894563727, 4.67740386295752647696242363002, 5.04122383286582937255022309137, 5.84936444694027856373721378685, 6.64444742424468432147445062151, 7.60461143813055641389754872174, 8.129146925387390203492825972053