| L(s) = 1 | − 2-s + 1.65·3-s + 4-s + 2.27·5-s − 1.65·6-s − 2.18·7-s − 8-s − 0.250·9-s − 2.27·10-s + 0.389·11-s + 1.65·12-s − 6.44·13-s + 2.18·14-s + 3.77·15-s + 16-s + 4.88·17-s + 0.250·18-s + 1.82·19-s + 2.27·20-s − 3.62·21-s − 0.389·22-s − 3.56·23-s − 1.65·24-s + 0.190·25-s + 6.44·26-s − 5.38·27-s − 2.18·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.957·3-s + 0.5·4-s + 1.01·5-s − 0.676·6-s − 0.825·7-s − 0.353·8-s − 0.0834·9-s − 0.720·10-s + 0.117·11-s + 0.478·12-s − 1.78·13-s + 0.583·14-s + 0.975·15-s + 0.250·16-s + 1.18·17-s + 0.0590·18-s + 0.418·19-s + 0.509·20-s − 0.790·21-s − 0.0831·22-s − 0.742·23-s − 0.338·24-s + 0.0381·25-s + 1.26·26-s − 1.03·27-s − 0.412·28-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 | \( 1 - 1.65T + 3T^{2} \) |
| 5 | \( 1 - 2.27T + 5T^{2} \) |
| 7 | \( 1 + 2.18T + 7T^{2} \) |
| 11 | \( 1 - 0.389T + 11T^{2} \) |
| 13 | \( 1 + 6.44T + 13T^{2} \) |
| 17 | \( 1 - 4.88T + 17T^{2} \) |
| 19 | \( 1 - 1.82T + 19T^{2} \) |
| 23 | \( 1 + 3.56T + 23T^{2} \) |
| 29 | \( 1 - 2.16T + 29T^{2} \) |
| 31 | \( 1 + 6.39T + 31T^{2} \) |
| 37 | \( 1 + 11.3T + 37T^{2} \) |
| 41 | \( 1 - 1.57T + 41T^{2} \) |
| 43 | \( 1 - 3.34T + 43T^{2} \) |
| 47 | \( 1 - 4.52T + 47T^{2} \) |
| 53 | \( 1 - 12.9T + 53T^{2} \) |
| 59 | \( 1 - 7.73T + 59T^{2} \) |
| 61 | \( 1 - 4.63T + 61T^{2} \) |
| 67 | \( 1 - 0.0366T + 67T^{2} \) |
| 71 | \( 1 + 7.83T + 71T^{2} \) |
| 73 | \( 1 + 12.1T + 73T^{2} \) |
| 79 | \( 1 + 13.5T + 79T^{2} \) |
| 83 | \( 1 + 14.4T + 83T^{2} \) |
| 89 | \( 1 + 12.5T + 89T^{2} \) |
| 97 | \( 1 + 10.0T + 97T^{2} \) |
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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
−8.225043559593015114960904963521, −7.29437663011100933299564843053, −7.01871319737111254511145730895, −5.63756928889891243694868085769, −5.56959258356456931197641766775, −3.99442581235636113209199141481, −3.02344593515506906458017434527, −2.48428096510741536613668941381, −1.62574330170056375554805070870, 0,
1.62574330170056375554805070870, 2.48428096510741536613668941381, 3.02344593515506906458017434527, 3.99442581235636113209199141481, 5.56959258356456931197641766775, 5.63756928889891243694868085769, 7.01871319737111254511145730895, 7.29437663011100933299564843053, 8.225043559593015114960904963521
