L(s) = 1 | + 2-s − 2.37·3-s + 4-s − 1.97·5-s − 2.37·6-s − 4.20·7-s + 8-s + 2.64·9-s − 1.97·10-s − 5.02·11-s − 2.37·12-s + 5.57·13-s − 4.20·14-s + 4.69·15-s + 16-s + 6.46·17-s + 2.64·18-s + 4.71·19-s − 1.97·20-s + 9.99·21-s − 5.02·22-s + 1.78·23-s − 2.37·24-s − 1.09·25-s + 5.57·26-s + 0.850·27-s − 4.20·28-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 1.37·3-s + 0.5·4-s − 0.884·5-s − 0.969·6-s − 1.58·7-s + 0.353·8-s + 0.880·9-s − 0.625·10-s − 1.51·11-s − 0.685·12-s + 1.54·13-s − 1.12·14-s + 1.21·15-s + 0.250·16-s + 1.56·17-s + 0.622·18-s + 1.08·19-s − 0.442·20-s + 2.18·21-s − 1.07·22-s + 0.372·23-s − 0.484·24-s − 0.218·25-s + 1.09·26-s + 0.163·27-s − 0.794·28-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 | \( 1 + 2.37T + 3T^{2} \) |
| 5 | \( 1 + 1.97T + 5T^{2} \) |
| 7 | \( 1 + 4.20T + 7T^{2} \) |
| 11 | \( 1 + 5.02T + 11T^{2} \) |
| 13 | \( 1 - 5.57T + 13T^{2} \) |
| 17 | \( 1 - 6.46T + 17T^{2} \) |
| 19 | \( 1 - 4.71T + 19T^{2} \) |
| 23 | \( 1 - 1.78T + 23T^{2} \) |
| 29 | \( 1 + 7.59T + 29T^{2} \) |
| 31 | \( 1 - 3.98T + 31T^{2} \) |
| 37 | \( 1 - 5.87T + 37T^{2} \) |
| 41 | \( 1 + 1.51T + 41T^{2} \) |
| 43 | \( 1 - 3.13T + 43T^{2} \) |
| 47 | \( 1 - 0.621T + 47T^{2} \) |
| 53 | \( 1 - 0.0835T + 53T^{2} \) |
| 59 | \( 1 - 0.667T + 59T^{2} \) |
| 61 | \( 1 + 11.5T + 61T^{2} \) |
| 67 | \( 1 + 12.5T + 67T^{2} \) |
| 71 | \( 1 + 2.09T + 71T^{2} \) |
| 73 | \( 1 + 11.2T + 73T^{2} \) |
| 79 | \( 1 + 15.5T + 79T^{2} \) |
| 83 | \( 1 - 1.87T + 83T^{2} \) |
| 89 | \( 1 - 8.22T + 89T^{2} \) |
| 97 | \( 1 + 15.0T + 97T^{2} \) |
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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
−7.55540065751776786232223092324, −7.43469587560505538296428775271, −6.14169632393828401063963243127, −5.93375470743037077738099462851, −5.30399833829285859479291254452, −4.30960922780176772525971007064, −3.36454040157008236017297734388, −3.01139098954918188212704874957, −1.07565008409424349464101738992, 0,
1.07565008409424349464101738992, 3.01139098954918188212704874957, 3.36454040157008236017297734388, 4.30960922780176772525971007064, 5.30399833829285859479291254452, 5.93375470743037077738099462851, 6.14169632393828401063963243127, 7.43469587560505538296428775271, 7.55540065751776786232223092324