| L(s) = 1 | + 2-s + 0.618·3-s + 4-s + 2·5-s + 0.618·6-s + 7-s + 8-s − 2.61·9-s + 2·10-s + 0.618·12-s + 1.23·13-s + 14-s + 1.23·15-s + 16-s − 0.854·17-s − 2.61·18-s + 6.85·19-s + 2·20-s + 0.618·21-s − 0.472·23-s + 0.618·24-s − 25-s + 1.23·26-s − 3.47·27-s + 28-s + 8·29-s + 1.23·30-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.356·3-s + 0.5·4-s + 0.894·5-s + 0.252·6-s + 0.377·7-s + 0.353·8-s − 0.872·9-s + 0.632·10-s + 0.178·12-s + 0.342·13-s + 0.267·14-s + 0.319·15-s + 0.250·16-s − 0.207·17-s − 0.617·18-s + 1.57·19-s + 0.447·20-s + 0.134·21-s − 0.0984·23-s + 0.126·24-s − 0.200·25-s + 0.242·26-s − 0.668·27-s + 0.188·28-s + 1.48·29-s + 0.225·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1694 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1694 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.674503010\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.674503010\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 \) |
| good | 3 | \( 1 - 0.618T + 3T^{2} \) |
| 5 | \( 1 - 2T + 5T^{2} \) |
| 13 | \( 1 - 1.23T + 13T^{2} \) |
| 17 | \( 1 + 0.854T + 17T^{2} \) |
| 19 | \( 1 - 6.85T + 19T^{2} \) |
| 23 | \( 1 + 0.472T + 23T^{2} \) |
| 29 | \( 1 - 8T + 29T^{2} \) |
| 31 | \( 1 - 4.47T + 31T^{2} \) |
| 37 | \( 1 - 5.70T + 37T^{2} \) |
| 41 | \( 1 - 5.85T + 41T^{2} \) |
| 43 | \( 1 + 1.85T + 43T^{2} \) |
| 47 | \( 1 + 11.7T + 47T^{2} \) |
| 53 | \( 1 + 12.9T + 53T^{2} \) |
| 59 | \( 1 - 10.8T + 59T^{2} \) |
| 61 | \( 1 - 2.47T + 61T^{2} \) |
| 67 | \( 1 + 10.0T + 67T^{2} \) |
| 71 | \( 1 - 1.52T + 71T^{2} \) |
| 73 | \( 1 - 4.61T + 73T^{2} \) |
| 79 | \( 1 + 3.23T + 79T^{2} \) |
| 83 | \( 1 + 16.0T + 83T^{2} \) |
| 89 | \( 1 - 0.326T + 89T^{2} \) |
| 97 | \( 1 + 9.85T + 97T^{2} \) |
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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
−9.490475665057443451206477421881, −8.418446235248361561700606744086, −7.86213403685719002325956655757, −6.72090902671777463030909493880, −5.97090961669993750380873693583, −5.32270441993476408220008134107, −4.43835411211271579983911229904, −3.20453525104208179384554795308, −2.54535825011063263019980920215, −1.33739753615855599136704493182,
1.33739753615855599136704493182, 2.54535825011063263019980920215, 3.20453525104208179384554795308, 4.43835411211271579983911229904, 5.32270441993476408220008134107, 5.97090961669993750380873693583, 6.72090902671777463030909493880, 7.86213403685719002325956655757, 8.418446235248361561700606744086, 9.490475665057443451206477421881
