| L(s) = 1 | − 2-s + 4-s − 5-s + 7-s − 8-s − 3·9-s + 10-s + 6·13-s − 14-s + 16-s − 2·17-s + 3·18-s − 20-s + 25-s − 6·26-s + 28-s − 6·29-s + 8·31-s − 32-s + 2·34-s − 35-s − 3·36-s − 10·37-s + 40-s − 2·41-s − 4·43-s + 3·45-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s − 0.447·5-s + 0.377·7-s − 0.353·8-s − 9-s + 0.316·10-s + 1.66·13-s − 0.267·14-s + 1/4·16-s − 0.485·17-s + 0.707·18-s − 0.223·20-s + 1/5·25-s − 1.17·26-s + 0.188·28-s − 1.11·29-s + 1.43·31-s − 0.176·32-s + 0.342·34-s − 0.169·35-s − 1/2·36-s − 1.64·37-s + 0.158·40-s − 0.312·41-s − 0.609·43-s + 0.447·45-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8470 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8470 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 \) |
| good | 3 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 6 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 + 8 T + p T^{2} \) |
| 61 | \( 1 - 14 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 + 16 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + 8 T + p T^{2} \) |
| 89 | \( 1 - 10 T + p T^{2} \) |
| 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
−17.34865696330956, −16.63280249078332, −16.05500888183225, −15.52735576701462, −15.05529262381296, −14.31888608298882, −13.65410840774739, −13.24195922976209, −12.22876614240727, −11.68307511294191, −11.27621041470977, −10.67469165206980, −10.18970420292551, −9.046331639181176, −8.780454827817629, −8.267889620477558, −7.616794734923788, −6.820567016120650, −6.128285639967756, −5.565486179673146, −4.618752397394013, −3.695169463770508, −3.091817928846595, −2.060782184560023, −1.140348346406679, 0,
1.140348346406679, 2.060782184560023, 3.091817928846595, 3.695169463770508, 4.618752397394013, 5.565486179673146, 6.128285639967756, 6.820567016120650, 7.616794734923788, 8.267889620477558, 8.780454827817629, 9.046331639181176, 10.18970420292551, 10.67469165206980, 11.27621041470977, 11.68307511294191, 12.22876614240727, 13.24195922976209, 13.65410840774739, 14.31888608298882, 15.05529262381296, 15.52735576701462, 16.05500888183225, 16.63280249078332, 17.34865696330956
