| L(s) = 1 | − 2.30·2-s − 2.77·3-s + 3.32·4-s + 2.46·5-s + 6.40·6-s − 0.00709·7-s − 3.05·8-s + 4.70·9-s − 5.69·10-s + 0.0942·11-s − 9.23·12-s + 2.65·13-s + 0.0163·14-s − 6.85·15-s + 0.408·16-s − 0.103·17-s − 10.8·18-s + 19-s + 8.20·20-s + 0.0197·21-s − 0.217·22-s + 6.73·23-s + 8.49·24-s + 1.08·25-s − 6.11·26-s − 4.73·27-s − 0.0236·28-s + ⋯ |
| L(s) = 1 | − 1.63·2-s − 1.60·3-s + 1.66·4-s + 1.10·5-s + 2.61·6-s − 0.00268·7-s − 1.08·8-s + 1.56·9-s − 1.80·10-s + 0.0284·11-s − 2.66·12-s + 0.735·13-s + 0.00437·14-s − 1.76·15-s + 0.102·16-s − 0.0249·17-s − 2.56·18-s + 0.229·19-s + 1.83·20-s + 0.00430·21-s − 0.0463·22-s + 1.40·23-s + 1.73·24-s + 0.217·25-s − 1.19·26-s − 0.912·27-s − 0.00446·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4009 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4009 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 19 | \( 1 - T \) |
| 211 | \( 1 - T \) |
| good | 2 | \( 1 + 2.30T + 2T^{2} \) |
| 3 | \( 1 + 2.77T + 3T^{2} \) |
| 5 | \( 1 - 2.46T + 5T^{2} \) |
| 7 | \( 1 + 0.00709T + 7T^{2} \) |
| 11 | \( 1 - 0.0942T + 11T^{2} \) |
| 13 | \( 1 - 2.65T + 13T^{2} \) |
| 17 | \( 1 + 0.103T + 17T^{2} \) |
| 23 | \( 1 - 6.73T + 23T^{2} \) |
| 29 | \( 1 - 1.19T + 29T^{2} \) |
| 31 | \( 1 + 3.37T + 31T^{2} \) |
| 37 | \( 1 + 2.20T + 37T^{2} \) |
| 41 | \( 1 + 4.74T + 41T^{2} \) |
| 43 | \( 1 - 0.284T + 43T^{2} \) |
| 47 | \( 1 + 5.74T + 47T^{2} \) |
| 53 | \( 1 + 8.09T + 53T^{2} \) |
| 59 | \( 1 + 15.0T + 59T^{2} \) |
| 61 | \( 1 - 6.40T + 61T^{2} \) |
| 67 | \( 1 + 12.4T + 67T^{2} \) |
| 71 | \( 1 - 5.05T + 71T^{2} \) |
| 73 | \( 1 - 8.19T + 73T^{2} \) |
| 79 | \( 1 + 13.2T + 79T^{2} \) |
| 83 | \( 1 - 2.51T + 83T^{2} \) |
| 89 | \( 1 + 14.9T + 89T^{2} \) |
| 97 | \( 1 - 13.5T + 97T^{2} \) |
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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
−8.198471695546204580589187295291, −7.24587469982929215808394551850, −6.61751249343114836430325468424, −6.12886723241708460854947793974, −5.36387927873060866017744582194, −4.63222225778285779483782834678, −3.09956128542251131771127780831, −1.73816247528810718291203690802, −1.20100227225711504002973467701, 0,
1.20100227225711504002973467701, 1.73816247528810718291203690802, 3.09956128542251131771127780831, 4.63222225778285779483782834678, 5.36387927873060866017744582194, 6.12886723241708460854947793974, 6.61751249343114836430325468424, 7.24587469982929215808394551850, 8.198471695546204580589187295291
