| L(s) = 1 | − 0.811·2-s − 3.20·3-s − 1.34·4-s + 2.59·6-s + 1.14·7-s + 2.71·8-s + 7.24·9-s + 3.86·11-s + 4.29·12-s + 3.57·13-s − 0.930·14-s + 0.478·16-s − 0.180·17-s − 5.88·18-s − 2.95·19-s − 3.66·21-s − 3.13·22-s + 0.568·23-s − 8.68·24-s − 2.89·26-s − 13.6·27-s − 1.53·28-s + 2.43·29-s + 8.50·31-s − 5.81·32-s − 12.3·33-s + 0.146·34-s + ⋯ |
| L(s) = 1 | − 0.574·2-s − 1.84·3-s − 0.670·4-s + 1.06·6-s + 0.433·7-s + 0.959·8-s + 2.41·9-s + 1.16·11-s + 1.23·12-s + 0.990·13-s − 0.248·14-s + 0.119·16-s − 0.0437·17-s − 1.38·18-s − 0.677·19-s − 0.800·21-s − 0.668·22-s + 0.118·23-s − 1.77·24-s − 0.568·26-s − 2.61·27-s − 0.290·28-s + 0.451·29-s + 1.52·31-s − 1.02·32-s − 2.15·33-s + 0.0251·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 \) |
| 241 | \( 1 - T \) |
| good | 2 | \( 1 + 0.811T + 2T^{2} \) |
| 3 | \( 1 + 3.20T + 3T^{2} \) |
| 7 | \( 1 - 1.14T + 7T^{2} \) |
| 11 | \( 1 - 3.86T + 11T^{2} \) |
| 13 | \( 1 - 3.57T + 13T^{2} \) |
| 17 | \( 1 + 0.180T + 17T^{2} \) |
| 19 | \( 1 + 2.95T + 19T^{2} \) |
| 23 | \( 1 - 0.568T + 23T^{2} \) |
| 29 | \( 1 - 2.43T + 29T^{2} \) |
| 31 | \( 1 - 8.50T + 31T^{2} \) |
| 37 | \( 1 + 5.08T + 37T^{2} \) |
| 41 | \( 1 + 0.552T + 41T^{2} \) |
| 43 | \( 1 + 2.81T + 43T^{2} \) |
| 47 | \( 1 + 11.6T + 47T^{2} \) |
| 53 | \( 1 + 6.54T + 53T^{2} \) |
| 59 | \( 1 - 1.20T + 59T^{2} \) |
| 61 | \( 1 - 5.06T + 61T^{2} \) |
| 67 | \( 1 + 8.79T + 67T^{2} \) |
| 71 | \( 1 + 3.53T + 71T^{2} \) |
| 73 | \( 1 - 0.894T + 73T^{2} \) |
| 79 | \( 1 + 1.87T + 79T^{2} \) |
| 83 | \( 1 + 0.339T + 83T^{2} \) |
| 89 | \( 1 + 7.76T + 89T^{2} \) |
| 97 | \( 1 - 0.485T + 97T^{2} \) |
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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
−7.78150028241917213164575775586, −6.62530823409730233796543886137, −6.51857623683891777485411766717, −5.59822722662109457337962130394, −4.77198932403995796458490040269, −4.41174430578026523282102469331, −3.55295192136758695630759708519, −1.59137696020202193016164312069, −1.12222168151911137602259799093, 0,
1.12222168151911137602259799093, 1.59137696020202193016164312069, 3.55295192136758695630759708519, 4.41174430578026523282102469331, 4.77198932403995796458490040269, 5.59822722662109457337962130394, 6.51857623683891777485411766717, 6.62530823409730233796543886137, 7.78150028241917213164575775586
