| L(s) = 1 | − 2-s − 3-s + 4-s + 6-s + 2·7-s − 8-s + 9-s + 11-s − 12-s − 2·14-s + 16-s − 17-s − 18-s − 5·19-s − 2·21-s − 22-s − 9·23-s + 24-s − 5·25-s − 27-s + 2·28-s − 5·29-s + 8·31-s − 32-s − 33-s + 34-s + 36-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.408·6-s + 0.755·7-s − 0.353·8-s + 1/3·9-s + 0.301·11-s − 0.288·12-s − 0.534·14-s + 1/4·16-s − 0.242·17-s − 0.235·18-s − 1.14·19-s − 0.436·21-s − 0.213·22-s − 1.87·23-s + 0.204·24-s − 25-s − 0.192·27-s + 0.377·28-s − 0.928·29-s + 1.43·31-s − 0.176·32-s − 0.174·33-s + 0.171·34-s + 1/6·36-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 189618 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 189618 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 3 | \( 1 + T \) |
| 11 | \( 1 - T \) |
| 13 | \( 1 \) |
| 17 | \( 1 + T \) |
| good | 5 | \( 1 + p T^{2} \) |
| 7 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + 5 T + p T^{2} \) |
| 23 | \( 1 + 9 T + p T^{2} \) |
| 29 | \( 1 + 5 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 + T + p T^{2} \) |
| 47 | \( 1 + 13 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 6 T + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 + T + p T^{2} \) |
| 73 | \( 1 + 16 T + p T^{2} \) |
| 79 | \( 1 + 12 T + p T^{2} \) |
| 83 | \( 1 + 17 T + p T^{2} \) |
| 89 | \( 1 + 3 T + p T^{2} \) |
| 97 | \( 1 - T + p T^{2} \) |
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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
−13.44423188495784, −13.09756756318677, −12.52513737047938, −11.97992003458327, −11.54028967126780, −11.29198789645661, −10.87935883442595, −10.10048577687131, −9.894983520726100, −9.520547690221538, −8.724413491155361, −8.210756156999241, −8.002510040968880, −7.516172375352143, −6.720811557681328, −6.382405989236890, −5.935709023789372, −5.444320145400147, −4.656132328334933, −4.212738215371121, −3.862756906761278, −2.846705324930367, −2.293704001059260, −1.605435372342959, −1.312224820501092, 0, 0,
1.312224820501092, 1.605435372342959, 2.293704001059260, 2.846705324930367, 3.862756906761278, 4.212738215371121, 4.656132328334933, 5.444320145400147, 5.935709023789372, 6.382405989236890, 6.720811557681328, 7.516172375352143, 8.002510040968880, 8.210756156999241, 8.724413491155361, 9.520547690221538, 9.894983520726100, 10.10048577687131, 10.87935883442595, 11.29198789645661, 11.54028967126780, 11.97992003458327, 12.52513737047938, 13.09756756318677, 13.44423188495784
