| L(s) = 1 | − 2.78·2-s − 2.78·3-s + 5.76·4-s − 2.35·5-s + 7.75·6-s + 2.45·7-s − 10.4·8-s + 4.73·9-s + 6.57·10-s − 0.138·11-s − 16.0·12-s − 0.680·13-s − 6.84·14-s + 6.56·15-s + 17.6·16-s − 1.50·17-s − 13.2·18-s + 1.47·19-s − 13.5·20-s − 6.83·21-s + 0.385·22-s + 1.82·23-s + 29.1·24-s + 0.567·25-s + 1.89·26-s − 4.83·27-s + 14.1·28-s + ⋯ |
| L(s) = 1 | − 1.97·2-s − 1.60·3-s + 2.88·4-s − 1.05·5-s + 3.16·6-s + 0.928·7-s − 3.70·8-s + 1.57·9-s + 2.07·10-s − 0.0417·11-s − 4.62·12-s − 0.188·13-s − 1.82·14-s + 1.69·15-s + 4.42·16-s − 0.364·17-s − 3.11·18-s + 0.337·19-s − 3.04·20-s − 1.49·21-s + 0.0822·22-s + 0.380·23-s + 5.95·24-s + 0.113·25-s + 0.371·26-s − 0.930·27-s + 2.67·28-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 | \( 1 + 2.78T + 2T^{2} \) |
| 3 | \( 1 + 2.78T + 3T^{2} \) |
| 5 | \( 1 + 2.35T + 5T^{2} \) |
| 7 | \( 1 - 2.45T + 7T^{2} \) |
| 11 | \( 1 + 0.138T + 11T^{2} \) |
| 13 | \( 1 + 0.680T + 13T^{2} \) |
| 17 | \( 1 + 1.50T + 17T^{2} \) |
| 19 | \( 1 - 1.47T + 19T^{2} \) |
| 23 | \( 1 - 1.82T + 23T^{2} \) |
| 29 | \( 1 - 1.90T + 29T^{2} \) |
| 31 | \( 1 - 2.55T + 31T^{2} \) |
| 37 | \( 1 - 3.62T + 37T^{2} \) |
| 41 | \( 1 - 0.248T + 41T^{2} \) |
| 43 | \( 1 + 9.11T + 43T^{2} \) |
| 47 | \( 1 + 5.29T + 47T^{2} \) |
| 53 | \( 1 - 0.0647T + 53T^{2} \) |
| 59 | \( 1 - 3.82T + 59T^{2} \) |
| 61 | \( 1 + 2.71T + 61T^{2} \) |
| 67 | \( 1 + 10.4T + 67T^{2} \) |
| 71 | \( 1 + 9.64T + 71T^{2} \) |
| 73 | \( 1 - 2.70T + 73T^{2} \) |
| 79 | \( 1 - 15.7T + 79T^{2} \) |
| 83 | \( 1 + 8.01T + 83T^{2} \) |
| 89 | \( 1 - 3.06T + 89T^{2} \) |
| 97 | \( 1 + 3.37T + 97T^{2} \) |
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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
−8.030284609463937743856338958144, −7.52696511127064374586223157298, −6.81871950744192008265581482594, −6.22705095329180300607502893408, −5.29798429378542738134060155280, −4.47043002729350533021843696806, −3.13833873348189337687795067107, −1.81536460738528462137785847780, −0.901559725427037563290415525167, 0,
0.901559725427037563290415525167, 1.81536460738528462137785847780, 3.13833873348189337687795067107, 4.47043002729350533021843696806, 5.29798429378542738134060155280, 6.22705095329180300607502893408, 6.81871950744192008265581482594, 7.52696511127064374586223157298, 8.030284609463937743856338958144
