| L(s) = 1 | + 3-s + 2·5-s + 7-s + 9-s − 4·11-s − 6·13-s + 2·15-s + 6·17-s − 19-s + 21-s − 8·23-s − 25-s + 27-s − 6·29-s − 4·33-s + 2·35-s − 2·37-s − 6·39-s + 6·41-s + 4·43-s + 2·45-s − 4·47-s + 49-s + 6·51-s + 2·53-s − 8·55-s − 57-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.894·5-s + 0.377·7-s + 1/3·9-s − 1.20·11-s − 1.66·13-s + 0.516·15-s + 1.45·17-s − 0.229·19-s + 0.218·21-s − 1.66·23-s − 1/5·25-s + 0.192·27-s − 1.11·29-s − 0.696·33-s + 0.338·35-s − 0.328·37-s − 0.960·39-s + 0.937·41-s + 0.609·43-s + 0.298·45-s − 0.583·47-s + 1/7·49-s + 0.840·51-s + 0.274·53-s − 1.07·55-s − 0.132·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6384 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| 19 | \( 1 + T \) |
| good | 5 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + 4 T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 - 12 T + p T^{2} \) |
| 83 | \( 1 + 16 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 + 10 T + p T^{2} \) |
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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
−7.81289041290558503789391755132, −7.25413178554646885558603733200, −6.07199882817985407849232206627, −5.54247416903438776389767114385, −4.90592679186157244858922128073, −4.02091490235343924056457356357, −2.95918278082195828665957653097, −2.30308408075194485728999765743, −1.64153172745758622418996190172, 0,
1.64153172745758622418996190172, 2.30308408075194485728999765743, 2.95918278082195828665957653097, 4.02091490235343924056457356357, 4.90592679186157244858922128073, 5.54247416903438776389767114385, 6.07199882817985407849232206627, 7.25413178554646885558603733200, 7.81289041290558503789391755132
