| L(s) = 1 | + 2-s − 0.0802·3-s + 4-s − 5-s − 0.0802·6-s + 2.53·7-s + 8-s − 2.99·9-s − 10-s + 11-s − 0.0802·12-s + 5.53·13-s + 2.53·14-s + 0.0802·15-s + 16-s − 1.79·17-s − 2.99·18-s − 1.35·19-s − 20-s − 0.203·21-s + 22-s + 2.94·23-s − 0.0802·24-s + 25-s + 5.53·26-s + 0.481·27-s + 2.53·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.0463·3-s + 0.5·4-s − 0.447·5-s − 0.0327·6-s + 0.957·7-s + 0.353·8-s − 0.997·9-s − 0.316·10-s + 0.301·11-s − 0.0231·12-s + 1.53·13-s + 0.676·14-s + 0.0207·15-s + 0.250·16-s − 0.434·17-s − 0.705·18-s − 0.310·19-s − 0.223·20-s − 0.0443·21-s + 0.213·22-s + 0.613·23-s − 0.0163·24-s + 0.200·25-s + 1.08·26-s + 0.0926·27-s + 0.478·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4730 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4730 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.246310745\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.246310745\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 11 | \( 1 - T \) |
| 43 | \( 1 - T \) |
| good | 3 | \( 1 + 0.0802T + 3T^{2} \) |
| 7 | \( 1 - 2.53T + 7T^{2} \) |
| 13 | \( 1 - 5.53T + 13T^{2} \) |
| 17 | \( 1 + 1.79T + 17T^{2} \) |
| 19 | \( 1 + 1.35T + 19T^{2} \) |
| 23 | \( 1 - 2.94T + 23T^{2} \) |
| 29 | \( 1 - 5.51T + 29T^{2} \) |
| 31 | \( 1 + 4.14T + 31T^{2} \) |
| 37 | \( 1 - 11.9T + 37T^{2} \) |
| 41 | \( 1 + 8.42T + 41T^{2} \) |
| 47 | \( 1 + 5.26T + 47T^{2} \) |
| 53 | \( 1 - 3.73T + 53T^{2} \) |
| 59 | \( 1 + 6.86T + 59T^{2} \) |
| 61 | \( 1 - 3.71T + 61T^{2} \) |
| 67 | \( 1 - 11.0T + 67T^{2} \) |
| 71 | \( 1 + 9.33T + 71T^{2} \) |
| 73 | \( 1 - 14.5T + 73T^{2} \) |
| 79 | \( 1 - 13.5T + 79T^{2} \) |
| 83 | \( 1 - 2.07T + 83T^{2} \) |
| 89 | \( 1 + 4.41T + 89T^{2} \) |
| 97 | \( 1 + 11.3T + 97T^{2} \) |
| 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
−8.434772266774032483003008727185, −7.61554289919810788162497342689, −6.59545508234487100497309592351, −6.14488134874067480364081795516, −5.24784548048829018345684162443, −4.61060772745756198318619197341, −3.79378252153628085429313305665, −3.07633782076056600436026148060, −2.02611504473269030462701626359, −0.933345278611256720657123980446,
0.933345278611256720657123980446, 2.02611504473269030462701626359, 3.07633782076056600436026148060, 3.79378252153628085429313305665, 4.61060772745756198318619197341, 5.24784548048829018345684162443, 6.14488134874067480364081795516, 6.59545508234487100497309592351, 7.61554289919810788162497342689, 8.434772266774032483003008727185
