| L(s) = 1 | + 2-s + 3-s + 4-s − 3·5-s + 6-s − 7-s + 8-s + 9-s − 3·10-s + 6·11-s + 12-s + 13-s − 14-s − 3·15-s + 16-s − 3·17-s + 18-s + 2·19-s − 3·20-s − 21-s + 6·22-s + 24-s + 4·25-s + 26-s + 27-s − 28-s − 3·29-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 1/2·4-s − 1.34·5-s + 0.408·6-s − 0.377·7-s + 0.353·8-s + 1/3·9-s − 0.948·10-s + 1.80·11-s + 0.288·12-s + 0.277·13-s − 0.267·14-s − 0.774·15-s + 1/4·16-s − 0.727·17-s + 0.235·18-s + 0.458·19-s − 0.670·20-s − 0.218·21-s + 1.27·22-s + 0.204·24-s + 4/5·25-s + 0.196·26-s + 0.192·27-s − 0.188·28-s − 0.557·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 71058 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 71058 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.625097865\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.625097865\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 13 | \( 1 - T \) |
| 911 | \( 1 + T \) |
| good | 5 | \( 1 + 3 T + p T^{2} \) |
| 7 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 - 6 T + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 3 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 - 5 T + p T^{2} \) |
| 41 | \( 1 + 3 T + p T^{2} \) |
| 43 | \( 1 + 7 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 + 12 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 - 3 T + p T^{2} \) |
| 73 | \( 1 + 16 T + p T^{2} \) |
| 79 | \( 1 + 10 T + p T^{2} \) |
| 83 | \( 1 - 18 T + p T^{2} \) |
| 89 | \( 1 - 15 T + p T^{2} \) |
| 97 | \( 1 - 17 T + p T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−14.24734568423390, −13.55695635272430, −13.22648927484265, −12.60709939328898, −12.07758223521279, −11.76828957001725, −11.11450404733104, −11.02820133232489, −9.988547338416024, −9.503751541747471, −8.967746360888631, −8.528300552246470, −7.862624721562807, −7.388964558341185, −6.868133976505911, −6.401677092863249, −5.875747154140817, −4.894829273949457, −4.427949185215730, −3.906006088563652, −3.440139802540520, −3.141420937414807, −2.068861026475902, −1.472019532921817, −0.5446364892804807,
0.5446364892804807, 1.472019532921817, 2.068861026475902, 3.141420937414807, 3.440139802540520, 3.906006088563652, 4.427949185215730, 4.894829273949457, 5.875747154140817, 6.401677092863249, 6.868133976505911, 7.388964558341185, 7.862624721562807, 8.528300552246470, 8.967746360888631, 9.503751541747471, 9.988547338416024, 11.02820133232489, 11.11450404733104, 11.76828957001725, 12.07758223521279, 12.60709939328898, 13.22648927484265, 13.55695635272430, 14.24734568423390
