| L(s) = 1 | + 0.462·3-s + 2.86·5-s − 7-s − 2.78·9-s − 5.72·11-s + 1.32·15-s − 17-s − 2.92·19-s − 0.462·21-s + 0.796·23-s + 3.18·25-s − 2.67·27-s − 1.07·29-s − 2.86·31-s − 2.64·33-s − 2.86·35-s − 2.92·37-s − 10.5·41-s − 11.6·43-s − 7.97·45-s − 5.85·47-s + 49-s − 0.462·51-s + 9.78·53-s − 16.3·55-s − 1.35·57-s + 8.09·59-s + ⋯ |
| L(s) = 1 | + 0.267·3-s + 1.27·5-s − 0.377·7-s − 0.928·9-s − 1.72·11-s + 0.341·15-s − 0.242·17-s − 0.671·19-s − 0.100·21-s + 0.166·23-s + 0.636·25-s − 0.515·27-s − 0.199·29-s − 0.513·31-s − 0.460·33-s − 0.483·35-s − 0.480·37-s − 1.64·41-s − 1.77·43-s − 1.18·45-s − 0.853·47-s + 0.142·49-s − 0.0647·51-s + 1.34·53-s − 2.20·55-s − 0.179·57-s + 1.05·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1904 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1904 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 + T \) |
| 17 | \( 1 + T \) |
| good | 3 | \( 1 - 0.462T + 3T^{2} \) |
| 5 | \( 1 - 2.86T + 5T^{2} \) |
| 11 | \( 1 + 5.72T + 11T^{2} \) |
| 13 | \( 1 + 13T^{2} \) |
| 19 | \( 1 + 2.92T + 19T^{2} \) |
| 23 | \( 1 - 0.796T + 23T^{2} \) |
| 29 | \( 1 + 1.07T + 29T^{2} \) |
| 31 | \( 1 + 2.86T + 31T^{2} \) |
| 37 | \( 1 + 2.92T + 37T^{2} \) |
| 41 | \( 1 + 10.5T + 41T^{2} \) |
| 43 | \( 1 + 11.6T + 43T^{2} \) |
| 47 | \( 1 + 5.85T + 47T^{2} \) |
| 53 | \( 1 - 9.78T + 53T^{2} \) |
| 59 | \( 1 - 8.09T + 59T^{2} \) |
| 61 | \( 1 - 11.7T + 61T^{2} \) |
| 67 | \( 1 - 13.8T + 67T^{2} \) |
| 71 | \( 1 - 9.16T + 71T^{2} \) |
| 73 | \( 1 + 6.18T + 73T^{2} \) |
| 79 | \( 1 + 8T + 79T^{2} \) |
| 83 | \( 1 + 17.8T + 83T^{2} \) |
| 89 | \( 1 - 1.20T + 89T^{2} \) |
| 97 | \( 1 - 4.06T + 97T^{2} \) |
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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
−8.573186758291310578008271090905, −8.420861538605539450280535626324, −7.15670972614571563604606766310, −6.37955443424674527728166143691, −5.42945206776228320631667181750, −5.15576716399970425719434571391, −3.56891463428426713562520825859, −2.61711504428476638245585468727, −1.98043581593644227665291838762, 0,
1.98043581593644227665291838762, 2.61711504428476638245585468727, 3.56891463428426713562520825859, 5.15576716399970425719434571391, 5.42945206776228320631667181750, 6.37955443424674527728166143691, 7.15670972614571563604606766310, 8.420861538605539450280535626324, 8.573186758291310578008271090905
