| L(s) = 1 | − 1.86·2-s + 1.86·3-s + 1.46·4-s − 2.46·5-s − 3.46·6-s − 7-s + 8-s + 0.462·9-s + 4.58·10-s + 1.32·11-s + 2.72·12-s − 5.86·13-s + 1.86·14-s − 4.58·15-s − 4.78·16-s + 17-s − 0.860·18-s − 5.64·19-s − 3.60·20-s − 1.86·21-s − 2.46·22-s + 1.92·23-s + 1.86·24-s + 1.06·25-s + 10.9·26-s − 4.72·27-s − 1.46·28-s + ⋯ |
| L(s) = 1 | − 1.31·2-s + 1.07·3-s + 0.731·4-s − 1.10·5-s − 1.41·6-s − 0.377·7-s + 0.353·8-s + 0.154·9-s + 1.44·10-s + 0.399·11-s + 0.785·12-s − 1.62·13-s + 0.497·14-s − 1.18·15-s − 1.19·16-s + 0.242·17-s − 0.202·18-s − 1.29·19-s − 0.805·20-s − 0.406·21-s − 0.525·22-s + 0.401·23-s + 0.379·24-s + 0.212·25-s + 2.13·26-s − 0.908·27-s − 0.276·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 371 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 371 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 7 | \( 1 + T \) |
| 53 | \( 1 + T \) |
| good | 2 | \( 1 + 1.86T + 2T^{2} \) |
| 3 | \( 1 - 1.86T + 3T^{2} \) |
| 5 | \( 1 + 2.46T + 5T^{2} \) |
| 11 | \( 1 - 1.32T + 11T^{2} \) |
| 13 | \( 1 + 5.86T + 13T^{2} \) |
| 17 | \( 1 - T + 17T^{2} \) |
| 19 | \( 1 + 5.64T + 19T^{2} \) |
| 23 | \( 1 - 1.92T + 23T^{2} \) |
| 29 | \( 1 - 3.64T + 29T^{2} \) |
| 31 | \( 1 + 5.13T + 31T^{2} \) |
| 37 | \( 1 + 1.06T + 37T^{2} \) |
| 41 | \( 1 + 5.44T + 41T^{2} \) |
| 43 | \( 1 + 0.860T + 43T^{2} \) |
| 47 | \( 1 + 7.10T + 47T^{2} \) |
| 59 | \( 1 - 0.676T + 59T^{2} \) |
| 61 | \( 1 + 3.20T + 61T^{2} \) |
| 67 | \( 1 - 0.398T + 67T^{2} \) |
| 71 | \( 1 + 11.6T + 71T^{2} \) |
| 73 | \( 1 - 15.4T + 73T^{2} \) |
| 79 | \( 1 - 15.3T + 79T^{2} \) |
| 83 | \( 1 - 1.34T + 83T^{2} \) |
| 89 | \( 1 - 17.0T + 89T^{2} \) |
| 97 | \( 1 - 4.22T + 97T^{2} \) |
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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
−10.64204774784177870640392227584, −9.708728940629202524778130167798, −9.015847834634783150205195111438, −8.219800331450313055410109739622, −7.61597445092945073880744124779, −6.75303369848413933520681035018, −4.72032571272710995484790899268, −3.51981928451076717535553038846, −2.19218535021881005678312082686, 0,
2.19218535021881005678312082686, 3.51981928451076717535553038846, 4.72032571272710995484790899268, 6.75303369848413933520681035018, 7.61597445092945073880744124779, 8.219800331450313055410109739622, 9.015847834634783150205195111438, 9.708728940629202524778130167798, 10.64204774784177870640392227584
