| L(s) = 1 | + 2-s − 0.156·3-s + 4-s − 5-s − 0.156·6-s − 3.72·7-s + 8-s − 2.97·9-s − 10-s − 11-s − 0.156·12-s − 1.30·13-s − 3.72·14-s + 0.156·15-s + 16-s − 7.93·17-s − 2.97·18-s + 2.16·19-s − 20-s + 0.584·21-s − 22-s + 9.25·23-s − 0.156·24-s + 25-s − 1.30·26-s + 0.938·27-s − 3.72·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.0906·3-s + 0.5·4-s − 0.447·5-s − 0.0640·6-s − 1.40·7-s + 0.353·8-s − 0.991·9-s − 0.316·10-s − 0.301·11-s − 0.0453·12-s − 0.360·13-s − 0.995·14-s + 0.0405·15-s + 0.250·16-s − 1.92·17-s − 0.701·18-s + 0.495·19-s − 0.223·20-s + 0.127·21-s − 0.213·22-s + 1.93·23-s − 0.0320·24-s + 0.200·25-s − 0.255·26-s + 0.180·27-s − 0.703·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4730 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4730 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.441671168\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.441671168\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 5 | \( 1 + T \) |
| 11 | \( 1 + T \) |
| 43 | \( 1 + T \) |
| good | 3 | \( 1 + 0.156T + 3T^{2} \) |
| 7 | \( 1 + 3.72T + 7T^{2} \) |
| 13 | \( 1 + 1.30T + 13T^{2} \) |
| 17 | \( 1 + 7.93T + 17T^{2} \) |
| 19 | \( 1 - 2.16T + 19T^{2} \) |
| 23 | \( 1 - 9.25T + 23T^{2} \) |
| 29 | \( 1 - 4.20T + 29T^{2} \) |
| 31 | \( 1 - 0.795T + 31T^{2} \) |
| 37 | \( 1 + 5.60T + 37T^{2} \) |
| 41 | \( 1 - 9.50T + 41T^{2} \) |
| 47 | \( 1 + 5.20T + 47T^{2} \) |
| 53 | \( 1 + 0.389T + 53T^{2} \) |
| 59 | \( 1 + 8.61T + 59T^{2} \) |
| 61 | \( 1 + 7.23T + 61T^{2} \) |
| 67 | \( 1 - 6.08T + 67T^{2} \) |
| 71 | \( 1 - 11.0T + 71T^{2} \) |
| 73 | \( 1 - 15.1T + 73T^{2} \) |
| 79 | \( 1 - 12.0T + 79T^{2} \) |
| 83 | \( 1 + 1.11T + 83T^{2} \) |
| 89 | \( 1 - 16.0T + 89T^{2} \) |
| 97 | \( 1 + 6.93T + 97T^{2} \) |
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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
−8.271526093308670106765521526544, −7.34886543335676213406201229331, −6.58798711136298536827719275966, −6.32564449920077067620873232580, −5.19480478212954251318523557362, −4.71754790780934863535143728760, −3.61607629141837298727433486839, −2.99364611529322708622934590902, −2.37223917917479785828202579168, −0.56074957273132449981928138712,
0.56074957273132449981928138712, 2.37223917917479785828202579168, 2.99364611529322708622934590902, 3.61607629141837298727433486839, 4.71754790780934863535143728760, 5.19480478212954251318523557362, 6.32564449920077067620873232580, 6.58798711136298536827719275966, 7.34886543335676213406201229331, 8.271526093308670106765521526544
