L(s) = 1 | − 2-s − 3-s + 4-s − 5-s + 6-s − 7-s − 8-s + 9-s + 10-s − 12-s + 13-s + 14-s + 15-s + 16-s − 2·17-s − 18-s − 19-s − 20-s + 21-s − 4·23-s + 24-s + 25-s − 26-s − 27-s − 28-s + 2·29-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 − 0.288·12-s + 0.277·13-s + 0.267·14-s + 0.258·15-s + 1/4·16-s − 0.485·17-s − 0.235·18-s − 0.229·19-s − 0.223·20-s + 0.218·21-s − 0.834·23-s + 0.204·24-s + 1/5·25-s − 0.196·26-s − 0.192·27-s − 0.188·28-s + 0.371·29-s − 0.182·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 51870 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 51870 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 13 | \( 1 - T \) |
| 19 | \( 1 + T \) |
good | 11 | \( 1 + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 12 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 + 16 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 + 12 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 - 2 T + p T^{2} \) |
| 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
−15.23129923017624, −14.57482981782321, −14.00386222593217, −13.21180358568269, −13.02049145077217, −12.23775504287759, −11.66368348992839, −11.60467355652453, −10.78624922973190, −10.25135107855380, −10.04569594646279, −9.228659330137271, −8.762900143688256, −8.185717990906224, −7.734787344998618, −6.985524876026676, −6.593779357501614, −6.111598626859794, −5.444184179753545, −4.700827862788747, −4.185259831336759, −3.354456271099381, −2.877961489805849, −1.811845723186736, −1.355199662134473, 0, 0,
1.355199662134473, 1.811845723186736, 2.877961489805849, 3.354456271099381, 4.185259831336759, 4.700827862788747, 5.444184179753545, 6.111598626859794, 6.593779357501614, 6.985524876026676, 7.734787344998618, 8.185717990906224, 8.762900143688256, 9.228659330137271, 10.04569594646279, 10.25135107855380, 10.78624922973190, 11.60467355652453, 11.66368348992839, 12.23775504287759, 13.02049145077217, 13.21180358568269, 14.00386222593217, 14.57482981782321, 15.23129923017624