| L(s) = 1 | − 2-s + 1.76·3-s + 4-s − 0.379·5-s − 1.76·6-s − 0.00141·7-s − 8-s + 0.121·9-s + 0.379·10-s − 1.92·11-s + 1.76·12-s − 0.508·13-s + 0.00141·14-s − 0.670·15-s + 16-s − 3.60·17-s − 0.121·18-s + 5.47·19-s − 0.379·20-s − 0.00249·21-s + 1.92·22-s + 23-s − 1.76·24-s − 4.85·25-s + 0.508·26-s − 5.08·27-s − 0.00141·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1.02·3-s + 0.5·4-s − 0.169·5-s − 0.721·6-s − 0.000534·7-s − 0.353·8-s + 0.0406·9-s + 0.119·10-s − 0.580·11-s + 0.510·12-s − 0.141·13-s + 0.000378·14-s − 0.173·15-s + 0.250·16-s − 0.874·17-s − 0.0287·18-s + 1.25·19-s − 0.0848·20-s − 0.000545·21-s + 0.410·22-s + 0.208·23-s − 0.360·24-s − 0.971·25-s + 0.0997·26-s − 0.978·27-s − 0.000267·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6026 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6026 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 23 | \( 1 - T \) |
| 131 | \( 1 - T \) |
| good | 3 | \( 1 - 1.76T + 3T^{2} \) |
| 5 | \( 1 + 0.379T + 5T^{2} \) |
| 7 | \( 1 + 0.00141T + 7T^{2} \) |
| 11 | \( 1 + 1.92T + 11T^{2} \) |
| 13 | \( 1 + 0.508T + 13T^{2} \) |
| 17 | \( 1 + 3.60T + 17T^{2} \) |
| 19 | \( 1 - 5.47T + 19T^{2} \) |
| 29 | \( 1 - 1.18T + 29T^{2} \) |
| 31 | \( 1 - 7.55T + 31T^{2} \) |
| 37 | \( 1 + 0.946T + 37T^{2} \) |
| 41 | \( 1 - 7.34T + 41T^{2} \) |
| 43 | \( 1 + 2.24T + 43T^{2} \) |
| 47 | \( 1 - 0.668T + 47T^{2} \) |
| 53 | \( 1 - 0.305T + 53T^{2} \) |
| 59 | \( 1 - 8.06T + 59T^{2} \) |
| 61 | \( 1 + 12.9T + 61T^{2} \) |
| 67 | \( 1 - 4.52T + 67T^{2} \) |
| 71 | \( 1 + 2.54T + 71T^{2} \) |
| 73 | \( 1 + 10.5T + 73T^{2} \) |
| 79 | \( 1 + 14.5T + 79T^{2} \) |
| 83 | \( 1 - 9.23T + 83T^{2} \) |
| 89 | \( 1 + 5.09T + 89T^{2} \) |
| 97 | \( 1 + 18.6T + 97T^{2} \) |
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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
−7.74366150365599769559506607934, −7.43923113787400139159111624984, −6.46632989069912072069076894401, −5.69270953401072059964088097958, −4.77143776834611868883627174195, −3.83074588575718909782312122515, −2.91172990356652271556074942044, −2.47504701496631892775441663118, −1.37670636111020642304279110276, 0,
1.37670636111020642304279110276, 2.47504701496631892775441663118, 2.91172990356652271556074942044, 3.83074588575718909782312122515, 4.77143776834611868883627174195, 5.69270953401072059964088097958, 6.46632989069912072069076894401, 7.43923113787400139159111624984, 7.74366150365599769559506607934
