| L(s) = 1 | − 2-s − 3-s + 4-s − 5-s + 6-s − 7-s − 8-s + 9-s + 10-s + 11-s − 12-s − 5·13-s + 14-s + 15-s + 16-s − 3·17-s − 18-s − 20-s + 21-s − 22-s + 24-s − 4·25-s + 5·26-s − 27-s − 28-s − 3·29-s − 30-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.447·5-s + 0.408·6-s − 0.377·7-s − 0.353·8-s + 1/3·9-s + 0.316·10-s + 0.301·11-s − 0.288·12-s − 1.38·13-s + 0.267·14-s + 0.258·15-s + 1/4·16-s − 0.727·17-s − 0.235·18-s − 0.223·20-s + 0.218·21-s − 0.213·22-s + 0.204·24-s − 4/5·25-s + 0.980·26-s − 0.192·27-s − 0.188·28-s − 0.557·29-s − 0.182·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 34914 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 34914 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 23 | \( 1 \) |
| good | 5 | \( 1 + T + p T^{2} \) |
| 7 | \( 1 + T + p T^{2} \) |
| 13 | \( 1 + 5 T + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 3 T + p T^{2} \) |
| 31 | \( 1 + 7 T + p T^{2} \) |
| 37 | \( 1 - 3 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 + 6 T + p T^{2} \) |
| 47 | \( 1 - T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 + 4 T + p T^{2} \) |
| 67 | \( 1 - 15 T + p T^{2} \) |
| 71 | \( 1 - 9 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 - 3 T + p T^{2} \) |
| 83 | \( 1 + 2 T + p T^{2} \) |
| 89 | \( 1 - 12 T + p T^{2} \) |
| 97 | \( 1 - 18 T + p T^{2} \) |
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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
−15.42456657133973, −14.74964107741490, −14.39604638620832, −13.54165140846375, −12.88539252309706, −12.55285729623521, −11.92428812871116, −11.34265220783237, −11.19611471589313, −10.31720842577839, −9.847602274081019, −9.459746495835910, −8.856427634395605, −8.145399184093345, −7.567923093913587, −7.110052128542202, −6.585209081989605, −5.997039912199449, −5.212858086395381, −4.744795050731772, −3.854100757492705, −3.371924573393016, −2.281860063523608, −1.890461391333972, −0.6649109874716728, 0,
0.6649109874716728, 1.890461391333972, 2.281860063523608, 3.371924573393016, 3.854100757492705, 4.744795050731772, 5.212858086395381, 5.997039912199449, 6.585209081989605, 7.110052128542202, 7.567923093913587, 8.145399184093345, 8.856427634395605, 9.459746495835910, 9.847602274081019, 10.31720842577839, 11.19611471589313, 11.34265220783237, 11.92428812871116, 12.55285729623521, 12.88539252309706, 13.54165140846375, 14.39604638620832, 14.74964107741490, 15.42456657133973
