| L(s) = 1 | − 2·2-s + 3-s + 2·4-s + 5-s − 2·6-s − 7-s + 9-s − 2·10-s + 2·11-s + 2·12-s + 13-s + 2·14-s + 15-s − 4·16-s − 2·18-s + 3·19-s + 2·20-s − 21-s − 4·22-s + 9·23-s − 4·25-s − 2·26-s + 27-s − 2·28-s + 29-s − 2·30-s + 5·31-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 0.577·3-s + 4-s + 0.447·5-s − 0.816·6-s − 0.377·7-s + 1/3·9-s − 0.632·10-s + 0.603·11-s + 0.577·12-s + 0.277·13-s + 0.534·14-s + 0.258·15-s − 16-s − 0.471·18-s + 0.688·19-s + 0.447·20-s − 0.218·21-s − 0.852·22-s + 1.87·23-s − 4/5·25-s − 0.392·26-s + 0.192·27-s − 0.377·28-s + 0.185·29-s − 0.365·30-s + 0.898·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 78897 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 78897 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 13 | \( 1 - T \) |
| 17 | \( 1 \) |
| good | 2 | \( 1 + p T + p T^{2} \) |
| 5 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 - 3 T + p T^{2} \) |
| 23 | \( 1 - 9 T + p T^{2} \) |
| 29 | \( 1 - T + p T^{2} \) |
| 31 | \( 1 - 5 T + p T^{2} \) |
| 37 | \( 1 - 8 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 9 T + p T^{2} \) |
| 47 | \( 1 + 3 T + p T^{2} \) |
| 53 | \( 1 - 3 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 + 2 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 + 5 T + p T^{2} \) |
| 79 | \( 1 - 13 T + p T^{2} \) |
| 83 | \( 1 + 11 T + p T^{2} \) |
| 89 | \( 1 - T + p T^{2} \) |
| 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
−14.23264942045676, −13.63484649572240, −13.33389049287727, −13.00701294808864, −11.94887533986199, −11.79911865781411, −11.04111002444931, −10.61839846813760, −9.952625235004748, −9.687362316225416, −9.225436125294167, −8.823037442188943, −8.267731512115173, −7.839324255541507, −7.151414583799872, −6.793815373176323, −6.261975252480698, −5.540223855616030, −4.755197608321075, −4.275152957606893, −3.329204717120972, −2.939538298095163, −2.174679263816294, −1.364594489630118, −1.079518121757922, 0,
1.079518121757922, 1.364594489630118, 2.174679263816294, 2.939538298095163, 3.329204717120972, 4.275152957606893, 4.755197608321075, 5.540223855616030, 6.261975252480698, 6.793815373176323, 7.151414583799872, 7.839324255541507, 8.267731512115173, 8.823037442188943, 9.225436125294167, 9.687362316225416, 9.952625235004748, 10.61839846813760, 11.04111002444931, 11.79911865781411, 11.94887533986199, 13.00701294808864, 13.33389049287727, 13.63484649572240, 14.23264942045676
