| L(s) = 1 | − 2.43·2-s + 1.77·3-s + 3.92·4-s − 0.360·5-s − 4.33·6-s − 1.53·7-s − 4.68·8-s + 0.166·9-s + 0.878·10-s − 2.71·11-s + 6.98·12-s + 5.42·13-s + 3.73·14-s − 0.642·15-s + 3.56·16-s + 1.82·17-s − 0.405·18-s − 1.34·19-s − 1.41·20-s − 2.73·21-s + 6.61·22-s + 6.95·23-s − 8.34·24-s − 4.86·25-s − 13.1·26-s − 5.04·27-s − 6.02·28-s + ⋯ |
| L(s) = 1 | − 1.72·2-s + 1.02·3-s + 1.96·4-s − 0.161·5-s − 1.76·6-s − 0.579·7-s − 1.65·8-s + 0.0555·9-s + 0.277·10-s − 0.819·11-s + 2.01·12-s + 1.50·13-s + 0.998·14-s − 0.165·15-s + 0.890·16-s + 0.443·17-s − 0.0956·18-s − 0.307·19-s − 0.316·20-s − 0.595·21-s + 1.41·22-s + 1.45·23-s − 1.70·24-s − 0.973·25-s − 2.58·26-s − 0.970·27-s − 1.13·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4003 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4003 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 4003 | \( 1+O(T) \) |
| good | 2 | \( 1 + 2.43T + 2T^{2} \) |
| 3 | \( 1 - 1.77T + 3T^{2} \) |
| 5 | \( 1 + 0.360T + 5T^{2} \) |
| 7 | \( 1 + 1.53T + 7T^{2} \) |
| 11 | \( 1 + 2.71T + 11T^{2} \) |
| 13 | \( 1 - 5.42T + 13T^{2} \) |
| 17 | \( 1 - 1.82T + 17T^{2} \) |
| 19 | \( 1 + 1.34T + 19T^{2} \) |
| 23 | \( 1 - 6.95T + 23T^{2} \) |
| 29 | \( 1 + 6.05T + 29T^{2} \) |
| 31 | \( 1 - 3.54T + 31T^{2} \) |
| 37 | \( 1 + 1.69T + 37T^{2} \) |
| 41 | \( 1 + 5.25T + 41T^{2} \) |
| 43 | \( 1 + 3.53T + 43T^{2} \) |
| 47 | \( 1 - 0.427T + 47T^{2} \) |
| 53 | \( 1 - 2.88T + 53T^{2} \) |
| 59 | \( 1 + 13.7T + 59T^{2} \) |
| 61 | \( 1 - 8.87T + 61T^{2} \) |
| 67 | \( 1 - 9.16T + 67T^{2} \) |
| 71 | \( 1 - 5.39T + 71T^{2} \) |
| 73 | \( 1 - 11.6T + 73T^{2} \) |
| 79 | \( 1 + 2.01T + 79T^{2} \) |
| 83 | \( 1 + 6.98T + 83T^{2} \) |
| 89 | \( 1 - 1.05T + 89T^{2} \) |
| 97 | \( 1 + 0.00171T + 97T^{2} \) |
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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
−8.241132713346257959675411832627, −7.78050628050082155263698391272, −6.94730951851492120716641112590, −6.23980664967493216690737006547, −5.29705043794537927869759815340, −3.76789293133610924473951103323, −3.13999544729471293747162317450, −2.28965675185189816558299506137, −1.30894491536030671758317189299, 0,
1.30894491536030671758317189299, 2.28965675185189816558299506137, 3.13999544729471293747162317450, 3.76789293133610924473951103323, 5.29705043794537927869759815340, 6.23980664967493216690737006547, 6.94730951851492120716641112590, 7.78050628050082155263698391272, 8.241132713346257959675411832627
