L(s) = 1 | − 2-s + 1.84·3-s + 4-s + 1.47·5-s − 1.84·6-s + 1.09·7-s − 8-s + 0.390·9-s − 1.47·10-s − 0.101·11-s + 1.84·12-s − 3.54·13-s − 1.09·14-s + 2.71·15-s + 16-s − 3.92·17-s − 0.390·18-s − 4.99·19-s + 1.47·20-s + 2.01·21-s + 0.101·22-s − 6.77·23-s − 1.84·24-s − 2.83·25-s + 3.54·26-s − 4.80·27-s + 1.09·28-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 1.06·3-s + 0.5·4-s + 0.658·5-s − 0.751·6-s + 0.413·7-s − 0.353·8-s + 0.130·9-s − 0.465·10-s − 0.0305·11-s + 0.531·12-s − 0.981·13-s − 0.292·14-s + 0.700·15-s + 0.250·16-s − 0.951·17-s − 0.0919·18-s − 1.14·19-s + 0.329·20-s + 0.440·21-s + 0.0215·22-s − 1.41·23-s − 0.375·24-s − 0.566·25-s + 0.694·26-s − 0.924·27-s + 0.206·28-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 | \( 1 - 1.84T + 3T^{2} \) |
| 5 | \( 1 - 1.47T + 5T^{2} \) |
| 7 | \( 1 - 1.09T + 7T^{2} \) |
| 11 | \( 1 + 0.101T + 11T^{2} \) |
| 13 | \( 1 + 3.54T + 13T^{2} \) |
| 17 | \( 1 + 3.92T + 17T^{2} \) |
| 19 | \( 1 + 4.99T + 19T^{2} \) |
| 23 | \( 1 + 6.77T + 23T^{2} \) |
| 29 | \( 1 - 7.27T + 29T^{2} \) |
| 31 | \( 1 - 7.41T + 31T^{2} \) |
| 37 | \( 1 - 2.39T + 37T^{2} \) |
| 41 | \( 1 + 9.66T + 41T^{2} \) |
| 43 | \( 1 + 10.0T + 43T^{2} \) |
| 47 | \( 1 - 1.56T + 47T^{2} \) |
| 53 | \( 1 - 1.00T + 53T^{2} \) |
| 59 | \( 1 + 3.27T + 59T^{2} \) |
| 61 | \( 1 + 2.11T + 61T^{2} \) |
| 67 | \( 1 - 4.55T + 67T^{2} \) |
| 71 | \( 1 + 15.4T + 71T^{2} \) |
| 73 | \( 1 + 1.14T + 73T^{2} \) |
| 79 | \( 1 - 4.89T + 79T^{2} \) |
| 83 | \( 1 - 10.7T + 83T^{2} \) |
| 89 | \( 1 + 4.38T + 89T^{2} \) |
| 97 | \( 1 + 4.01T + 97T^{2} \) |
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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
−8.256738380996560508460972018078, −7.69565535942828271087430688261, −6.62141508819304850794127950529, −6.18495017721810002474698376357, −5.00123579707009791831052267993, −4.22771397617330078568819993630, −3.05082355926008325027804780994, −2.25656850252401365854856491925, −1.79222727132922896208324961463, 0,
1.79222727132922896208324961463, 2.25656850252401365854856491925, 3.05082355926008325027804780994, 4.22771397617330078568819993630, 5.00123579707009791831052267993, 6.18495017721810002474698376357, 6.62141508819304850794127950529, 7.69565535942828271087430688261, 8.256738380996560508460972018078