| L(s) = 1 | − 1.99·2-s − 3-s + 1.97·4-s − 1.27·5-s + 1.99·6-s + 0.558·7-s + 0.0544·8-s + 9-s + 2.54·10-s + 3.37·11-s − 1.97·12-s + 2.95·13-s − 1.11·14-s + 1.27·15-s − 4.05·16-s − 17-s − 1.99·18-s − 5.11·19-s − 2.52·20-s − 0.558·21-s − 6.72·22-s + 4.39·23-s − 0.0544·24-s − 3.36·25-s − 5.88·26-s − 27-s + 1.10·28-s + ⋯ |
| L(s) = 1 | − 1.40·2-s − 0.577·3-s + 0.986·4-s − 0.571·5-s + 0.813·6-s + 0.210·7-s + 0.0192·8-s + 0.333·9-s + 0.805·10-s + 1.01·11-s − 0.569·12-s + 0.818·13-s − 0.297·14-s + 0.330·15-s − 1.01·16-s − 0.242·17-s − 0.469·18-s − 1.17·19-s − 0.563·20-s − 0.121·21-s − 1.43·22-s + 0.915·23-s − 0.0111·24-s − 0.673·25-s − 1.15·26-s − 0.192·27-s + 0.208·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4029 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4029 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 + T \) |
| 17 | \( 1 + T \) |
| 79 | \( 1 + T \) |
| good | 2 | \( 1 + 1.99T + 2T^{2} \) |
| 5 | \( 1 + 1.27T + 5T^{2} \) |
| 7 | \( 1 - 0.558T + 7T^{2} \) |
| 11 | \( 1 - 3.37T + 11T^{2} \) |
| 13 | \( 1 - 2.95T + 13T^{2} \) |
| 19 | \( 1 + 5.11T + 19T^{2} \) |
| 23 | \( 1 - 4.39T + 23T^{2} \) |
| 29 | \( 1 + 0.447T + 29T^{2} \) |
| 31 | \( 1 - 10.1T + 31T^{2} \) |
| 37 | \( 1 + 9.43T + 37T^{2} \) |
| 41 | \( 1 + 8.62T + 41T^{2} \) |
| 43 | \( 1 + 1.04T + 43T^{2} \) |
| 47 | \( 1 - 0.300T + 47T^{2} \) |
| 53 | \( 1 + 10.9T + 53T^{2} \) |
| 59 | \( 1 - 9.17T + 59T^{2} \) |
| 61 | \( 1 + 11.2T + 61T^{2} \) |
| 67 | \( 1 - 10.3T + 67T^{2} \) |
| 71 | \( 1 - 4.59T + 71T^{2} \) |
| 73 | \( 1 + 1.53T + 73T^{2} \) |
| 83 | \( 1 + 8.23T + 83T^{2} \) |
| 89 | \( 1 - 13.0T + 89T^{2} \) |
| 97 | \( 1 + 4.43T + 97T^{2} \) |
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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
−8.397891364983673262514384906943, −7.48918108707689484954534995831, −6.59087290503275069227304769618, −6.39412503129592810282784293025, −5.00235654656756267025555540005, −4.30600517018787751942544359692, −3.41866468582641016681141388185, −1.93261495166814071903335875858, −1.13614621598442933519971844083, 0,
1.13614621598442933519971844083, 1.93261495166814071903335875858, 3.41866468582641016681141388185, 4.30600517018787751942544359692, 5.00235654656756267025555540005, 6.39412503129592810282784293025, 6.59087290503275069227304769618, 7.48918108707689484954534995831, 8.397891364983673262514384906943
