| L(s) = 1 | + 0.384·2-s + 3-s − 1.85·4-s + 0.384·6-s − 7-s − 1.48·8-s + 9-s − 0.436·11-s − 1.85·12-s − 4.23·13-s − 0.384·14-s + 3.13·16-s − 17-s + 0.384·18-s + 4.11·19-s − 21-s − 0.167·22-s + 2.68·23-s − 1.48·24-s − 1.63·26-s + 27-s + 1.85·28-s + 0.858·29-s − 5.36·31-s + 4.17·32-s − 0.436·33-s − 0.384·34-s + ⋯ |
| L(s) = 1 | + 0.272·2-s + 0.577·3-s − 0.925·4-s + 0.157·6-s − 0.377·7-s − 0.524·8-s + 0.333·9-s − 0.131·11-s − 0.534·12-s − 1.17·13-s − 0.102·14-s + 0.783·16-s − 0.242·17-s + 0.0907·18-s + 0.943·19-s − 0.218·21-s − 0.0357·22-s + 0.558·23-s − 0.302·24-s − 0.319·26-s + 0.192·27-s + 0.349·28-s + 0.159·29-s − 0.962·31-s + 0.737·32-s − 0.0759·33-s − 0.0660·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8925 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8925 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 - T \) |
| 5 | \( 1 \) |
| 7 | \( 1 + T \) |
| 17 | \( 1 + T \) |
| good | 2 | \( 1 - 0.384T + 2T^{2} \) |
| 11 | \( 1 + 0.436T + 11T^{2} \) |
| 13 | \( 1 + 4.23T + 13T^{2} \) |
| 19 | \( 1 - 4.11T + 19T^{2} \) |
| 23 | \( 1 - 2.68T + 23T^{2} \) |
| 29 | \( 1 - 0.858T + 29T^{2} \) |
| 31 | \( 1 + 5.36T + 31T^{2} \) |
| 37 | \( 1 - 8.60T + 37T^{2} \) |
| 41 | \( 1 + 4.62T + 41T^{2} \) |
| 43 | \( 1 - 8.44T + 43T^{2} \) |
| 47 | \( 1 - 5.75T + 47T^{2} \) |
| 53 | \( 1 + 5.04T + 53T^{2} \) |
| 59 | \( 1 + 0.802T + 59T^{2} \) |
| 61 | \( 1 + 13.6T + 61T^{2} \) |
| 67 | \( 1 - 5.61T + 67T^{2} \) |
| 71 | \( 1 - 14.3T + 71T^{2} \) |
| 73 | \( 1 + 11.2T + 73T^{2} \) |
| 79 | \( 1 - 9.28T + 79T^{2} \) |
| 83 | \( 1 + 12.3T + 83T^{2} \) |
| 89 | \( 1 + 9.31T + 89T^{2} \) |
| 97 | \( 1 + 14.6T + 97T^{2} \) |
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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
−7.51874123253474829492529050023, −6.82870567039406496927511629889, −5.86108599083390272877826540105, −5.22927805095063930420378190618, −4.55760177839598900478601426261, −3.89588124341252050946810529810, −3.05712726367101173412226194911, −2.49377218745075540295614475648, −1.17544535111514538512508848795, 0,
1.17544535111514538512508848795, 2.49377218745075540295614475648, 3.05712726367101173412226194911, 3.89588124341252050946810529810, 4.55760177839598900478601426261, 5.22927805095063930420378190618, 5.86108599083390272877826540105, 6.82870567039406496927511629889, 7.51874123253474829492529050023
