| L(s) = 1 | − 2.72·2-s + 2.89·3-s + 5.42·4-s − 2.27·5-s − 7.89·6-s − 3.84·7-s − 9.34·8-s + 5.38·9-s + 6.20·10-s + 2.32·11-s + 15.7·12-s + 0.203·13-s + 10.4·14-s − 6.59·15-s + 14.6·16-s + 1.55·17-s − 14.6·18-s − 0.522·19-s − 12.3·20-s − 11.1·21-s − 6.33·22-s − 2.89·23-s − 27.0·24-s + 0.190·25-s − 0.555·26-s + 6.89·27-s − 20.8·28-s + ⋯ |
| L(s) = 1 | − 1.92·2-s + 1.67·3-s + 2.71·4-s − 1.01·5-s − 3.22·6-s − 1.45·7-s − 3.30·8-s + 1.79·9-s + 1.96·10-s + 0.700·11-s + 4.53·12-s + 0.0565·13-s + 2.79·14-s − 1.70·15-s + 3.65·16-s + 0.377·17-s − 3.45·18-s − 0.119·19-s − 2.76·20-s − 2.42·21-s − 1.35·22-s − 0.604·23-s − 5.52·24-s + 0.0380·25-s − 0.109·26-s + 1.32·27-s − 3.94·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8003 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8003 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 53 | \( 1 + T \) |
| 151 | \( 1 + T \) |
| good | 2 | \( 1 + 2.72T + 2T^{2} \) |
| 3 | \( 1 - 2.89T + 3T^{2} \) |
| 5 | \( 1 + 2.27T + 5T^{2} \) |
| 7 | \( 1 + 3.84T + 7T^{2} \) |
| 11 | \( 1 - 2.32T + 11T^{2} \) |
| 13 | \( 1 - 0.203T + 13T^{2} \) |
| 17 | \( 1 - 1.55T + 17T^{2} \) |
| 19 | \( 1 + 0.522T + 19T^{2} \) |
| 23 | \( 1 + 2.89T + 23T^{2} \) |
| 29 | \( 1 - 1.67T + 29T^{2} \) |
| 31 | \( 1 + 6.73T + 31T^{2} \) |
| 37 | \( 1 - 2.76T + 37T^{2} \) |
| 41 | \( 1 - 10.0T + 41T^{2} \) |
| 43 | \( 1 - 3.67T + 43T^{2} \) |
| 47 | \( 1 + 0.655T + 47T^{2} \) |
| 59 | \( 1 + 0.383T + 59T^{2} \) |
| 61 | \( 1 - 5.23T + 61T^{2} \) |
| 67 | \( 1 + 5.10T + 67T^{2} \) |
| 71 | \( 1 + 6.43T + 71T^{2} \) |
| 73 | \( 1 - 0.0435T + 73T^{2} \) |
| 79 | \( 1 - 6.92T + 79T^{2} \) |
| 83 | \( 1 + 3.05T + 83T^{2} \) |
| 89 | \( 1 - 15.0T + 89T^{2} \) |
| 97 | \( 1 + 7.18T + 97T^{2} \) |
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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
−7.64332606041262910787401986801, −7.33202129539491195376833598217, −6.57703476962870076536728613416, −5.92565765481948654620862498248, −4.04131764498615179746719434268, −3.57309682535169173515940437058, −2.87860072904659180745374565443, −2.19977219813320955881505973569, −1.13016156296996711343366681411, 0,
1.13016156296996711343366681411, 2.19977219813320955881505973569, 2.87860072904659180745374565443, 3.57309682535169173515940437058, 4.04131764498615179746719434268, 5.92565765481948654620862498248, 6.57703476962870076536728613416, 7.33202129539491195376833598217, 7.64332606041262910787401986801
