| L(s) = 1 | + 0.618·2-s + 2.23·3-s − 1.61·4-s − 5-s + 1.38·6-s − 4.23·7-s − 2.23·8-s + 2.00·9-s − 0.618·10-s + 0.236·11-s − 3.61·12-s − 2.61·14-s − 2.23·15-s + 1.85·16-s − 3.47·17-s + 1.23·18-s − 4.23·19-s + 1.61·20-s − 9.47·21-s + 0.145·22-s + 3.76·23-s − 5.00·24-s + 25-s − 2.23·27-s + 6.85·28-s − 7.47·29-s − 1.38·30-s + ⋯ |
| L(s) = 1 | + 0.437·2-s + 1.29·3-s − 0.809·4-s − 0.447·5-s + 0.564·6-s − 1.60·7-s − 0.790·8-s + 0.666·9-s − 0.195·10-s + 0.0711·11-s − 1.04·12-s − 0.699·14-s − 0.577·15-s + 0.463·16-s − 0.842·17-s + 0.291·18-s − 0.971·19-s + 0.361·20-s − 2.06·21-s + 0.0311·22-s + 0.784·23-s − 1.02·24-s + 0.200·25-s − 0.430·27-s + 1.29·28-s − 1.38·29-s − 0.252·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 845 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 845 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 + T \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 - 0.618T + 2T^{2} \) |
| 3 | \( 1 - 2.23T + 3T^{2} \) |
| 7 | \( 1 + 4.23T + 7T^{2} \) |
| 11 | \( 1 - 0.236T + 11T^{2} \) |
| 17 | \( 1 + 3.47T + 17T^{2} \) |
| 19 | \( 1 + 4.23T + 19T^{2} \) |
| 23 | \( 1 - 3.76T + 23T^{2} \) |
| 29 | \( 1 + 7.47T + 29T^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 + 3T + 37T^{2} \) |
| 41 | \( 1 + 11.9T + 41T^{2} \) |
| 43 | \( 1 + 6.23T + 43T^{2} \) |
| 47 | \( 1 - 4.94T + 47T^{2} \) |
| 53 | \( 1 - 6T + 53T^{2} \) |
| 59 | \( 1 - 0.708T + 59T^{2} \) |
| 61 | \( 1 - 14.4T + 61T^{2} \) |
| 67 | \( 1 - 2.70T + 67T^{2} \) |
| 71 | \( 1 - 6.23T + 71T^{2} \) |
| 73 | \( 1 - 6T + 73T^{2} \) |
| 79 | \( 1 + 79T^{2} \) |
| 83 | \( 1 + 8.94T + 83T^{2} \) |
| 89 | \( 1 - 9T + 89T^{2} \) |
| 97 | \( 1 - 3.47T + 97T^{2} \) |
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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
−9.543327251904744593915378526396, −8.849061319706564619488413156442, −8.438947367349405128549731943184, −7.18310541081407035948449286458, −6.38227455954925179501878282735, −5.15525107464586761210442922481, −3.83728644848747355831933995873, −3.52735459674842479947563113735, −2.43367768257096728715119096413, 0,
2.43367768257096728715119096413, 3.52735459674842479947563113735, 3.83728644848747355831933995873, 5.15525107464586761210442922481, 6.38227455954925179501878282735, 7.18310541081407035948449286458, 8.438947367349405128549731943184, 8.849061319706564619488413156442, 9.543327251904744593915378526396
