| L(s) = 1 | + 2-s − 2·3-s + 4-s + 0.732·5-s − 2·6-s + 8-s + 9-s + 0.732·10-s + 0.732·11-s − 2·12-s − 1.46·13-s − 1.46·15-s + 16-s + 18-s − 3.46·19-s + 0.732·20-s + 0.732·22-s − 2·23-s − 2·24-s − 4.46·25-s − 1.46·26-s + 4·27-s + 9.66·29-s − 1.46·30-s + 5.46·31-s + 32-s − 1.46·33-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1.15·3-s + 0.5·4-s + 0.327·5-s − 0.816·6-s + 0.353·8-s + 0.333·9-s + 0.231·10-s + 0.220·11-s − 0.577·12-s − 0.406·13-s − 0.378·15-s + 0.250·16-s + 0.235·18-s − 0.794·19-s + 0.163·20-s + 0.156·22-s − 0.417·23-s − 0.408·24-s − 0.892·25-s − 0.287·26-s + 0.769·27-s + 1.79·29-s − 0.267·30-s + 0.981·31-s + 0.176·32-s − 0.254·33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4018 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4018 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 7 | \( 1 \) |
| 41 | \( 1 + T \) |
| good | 3 | \( 1 + 2T + 3T^{2} \) |
| 5 | \( 1 - 0.732T + 5T^{2} \) |
| 11 | \( 1 - 0.732T + 11T^{2} \) |
| 13 | \( 1 + 1.46T + 13T^{2} \) |
| 17 | \( 1 + 17T^{2} \) |
| 19 | \( 1 + 3.46T + 19T^{2} \) |
| 23 | \( 1 + 2T + 23T^{2} \) |
| 29 | \( 1 - 9.66T + 29T^{2} \) |
| 31 | \( 1 - 5.46T + 31T^{2} \) |
| 37 | \( 1 + 8.92T + 37T^{2} \) |
| 43 | \( 1 + 8.39T + 43T^{2} \) |
| 47 | \( 1 + 7.46T + 47T^{2} \) |
| 53 | \( 1 + 9.66T + 53T^{2} \) |
| 59 | \( 1 - 11.1T + 59T^{2} \) |
| 61 | \( 1 + 3.26T + 61T^{2} \) |
| 67 | \( 1 + 10.1T + 67T^{2} \) |
| 71 | \( 1 - 9.46T + 71T^{2} \) |
| 73 | \( 1 + 4.92T + 73T^{2} \) |
| 79 | \( 1 + 79T^{2} \) |
| 83 | \( 1 + 1.26T + 83T^{2} \) |
| 89 | \( 1 - 0.928T + 89T^{2} \) |
| 97 | \( 1 + 2T + 97T^{2} \) |
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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
−8.053190726737274642666624697475, −6.89721619942946295496946347694, −6.44532218084505847384035071756, −5.89484845256824010143166941591, −4.99673245697035806642462068316, −4.62012306950026671402136918281, −3.53110984272985192226646014569, −2.52062016436486131010692971363, −1.44945425742225656385051570870, 0,
1.44945425742225656385051570870, 2.52062016436486131010692971363, 3.53110984272985192226646014569, 4.62012306950026671402136918281, 4.99673245697035806642462068316, 5.89484845256824010143166941591, 6.44532218084505847384035071756, 6.89721619942946295496946347694, 8.053190726737274642666624697475
