| L(s) = 1 | + 2-s + 0.926·3-s + 4-s + 3.02·5-s + 0.926·6-s − 4.27·7-s + 8-s − 2.14·9-s + 3.02·10-s − 3.02·11-s + 0.926·12-s − 4.36·13-s − 4.27·14-s + 2.80·15-s + 16-s + 6.53·17-s − 2.14·18-s − 2.89·19-s + 3.02·20-s − 3.96·21-s − 3.02·22-s − 6.78·23-s + 0.926·24-s + 4.17·25-s − 4.36·26-s − 4.76·27-s − 4.27·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.534·3-s + 0.5·4-s + 1.35·5-s + 0.378·6-s − 1.61·7-s + 0.353·8-s − 0.713·9-s + 0.957·10-s − 0.913·11-s + 0.267·12-s − 1.21·13-s − 1.14·14-s + 0.724·15-s + 0.250·16-s + 1.58·17-s − 0.504·18-s − 0.663·19-s + 0.677·20-s − 0.864·21-s − 0.645·22-s − 1.41·23-s + 0.189·24-s + 0.834·25-s − 0.855·26-s − 0.916·27-s − 0.807·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4022 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4022 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 2011 | \( 1 - T \) |
| good | 3 | \( 1 - 0.926T + 3T^{2} \) |
| 5 | \( 1 - 3.02T + 5T^{2} \) |
| 7 | \( 1 + 4.27T + 7T^{2} \) |
| 11 | \( 1 + 3.02T + 11T^{2} \) |
| 13 | \( 1 + 4.36T + 13T^{2} \) |
| 17 | \( 1 - 6.53T + 17T^{2} \) |
| 19 | \( 1 + 2.89T + 19T^{2} \) |
| 23 | \( 1 + 6.78T + 23T^{2} \) |
| 29 | \( 1 + 0.521T + 29T^{2} \) |
| 31 | \( 1 + 4.08T + 31T^{2} \) |
| 37 | \( 1 + 1.59T + 37T^{2} \) |
| 41 | \( 1 - 1.34T + 41T^{2} \) |
| 43 | \( 1 - 0.530T + 43T^{2} \) |
| 47 | \( 1 - 8.92T + 47T^{2} \) |
| 53 | \( 1 + 1.15T + 53T^{2} \) |
| 59 | \( 1 + 1.38T + 59T^{2} \) |
| 61 | \( 1 - 2.43T + 61T^{2} \) |
| 67 | \( 1 + 14.7T + 67T^{2} \) |
| 71 | \( 1 + 9.92T + 71T^{2} \) |
| 73 | \( 1 + 8.02T + 73T^{2} \) |
| 79 | \( 1 - 1.91T + 79T^{2} \) |
| 83 | \( 1 + 5.45T + 83T^{2} \) |
| 89 | \( 1 - 3.78T + 89T^{2} \) |
| 97 | \( 1 + 2.07T + 97T^{2} \) |
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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
−7.913646787267161156418059964348, −7.30609108605165207161125784260, −6.34683419799724479970145626075, −5.66171262856763897138642456008, −5.50014315631561595439198237414, −4.14979661903306801373476827151, −3.12588527697088474637594380910, −2.69809288688178474756154281378, −1.93967349923444076380360067817, 0,
1.93967349923444076380360067817, 2.69809288688178474756154281378, 3.12588527697088474637594380910, 4.14979661903306801373476827151, 5.50014315631561595439198237414, 5.66171262856763897138642456008, 6.34683419799724479970145626075, 7.30609108605165207161125784260, 7.913646787267161156418059964348
