| L(s) = 1 | − 2-s + 3-s − 4-s − 2·5-s − 6-s + 7-s + 3·8-s + 9-s + 2·10-s + 11-s − 12-s − 6·13-s − 14-s − 2·15-s − 16-s − 2·17-s − 18-s + 4·19-s + 2·20-s + 21-s − 22-s + 3·24-s − 25-s + 6·26-s + 27-s − 28-s + 2·29-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s − 1/2·4-s − 0.894·5-s − 0.408·6-s + 0.377·7-s + 1.06·8-s + 1/3·9-s + 0.632·10-s + 0.301·11-s − 0.288·12-s − 1.66·13-s − 0.267·14-s − 0.516·15-s − 1/4·16-s − 0.485·17-s − 0.235·18-s + 0.917·19-s + 0.447·20-s + 0.218·21-s − 0.213·22-s + 0.612·24-s − 1/5·25-s + 1.17·26-s + 0.192·27-s − 0.188·28-s + 0.371·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 221991 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 221991 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 - T \) |
| 31 | \( 1 \) |
| good | 2 | \( 1 + T + p T^{2} \) |
| 5 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 + 16 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 + 18 T + p T^{2} \) |
| 97 | \( 1 - 2 T + p T^{2} \) |
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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
−13.17198468607955, −12.68763226311970, −12.25158053860225, −11.80779716677468, −11.24638582355857, −10.91822045922006, −10.12135107140502, −9.811690514148558, −9.531119688989177, −8.840262576720845, −8.566021416492797, −7.996511130713923, −7.634449794281135, −7.143268763227168, −7.000703321035663, −5.944452272039397, −5.350758594976821, −4.751881361843243, −4.393411366764971, −4.000708625120081, −3.251410422562819, −2.742238183351036, −2.007054391990224, −1.428937710243023, −0.6413442232704574, 0,
0.6413442232704574, 1.428937710243023, 2.007054391990224, 2.742238183351036, 3.251410422562819, 4.000708625120081, 4.393411366764971, 4.751881361843243, 5.350758594976821, 5.944452272039397, 7.000703321035663, 7.143268763227168, 7.634449794281135, 7.996511130713923, 8.566021416492797, 8.840262576720845, 9.531119688989177, 9.811690514148558, 10.12135107140502, 10.91822045922006, 11.24638582355857, 11.80779716677468, 12.25158053860225, 12.68763226311970, 13.17198468607955
