L(s) = 1 | − 2-s + 3-s + 4-s − 0.621·5-s − 6-s + 2.55·7-s − 8-s + 9-s + 0.621·10-s + 0.810·11-s + 12-s − 0.218·13-s − 2.55·14-s − 0.621·15-s + 16-s + 17-s − 18-s − 7.58·19-s − 0.621·20-s + 2.55·21-s − 0.810·22-s − 3.76·23-s − 24-s − 4.61·25-s + 0.218·26-s + 27-s + 2.55·28-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 0.577·3-s + 0.5·4-s − 0.277·5-s − 0.408·6-s + 0.964·7-s − 0.353·8-s + 0.333·9-s + 0.196·10-s + 0.244·11-s + 0.288·12-s − 0.0605·13-s − 0.682·14-s − 0.160·15-s + 0.250·16-s + 0.242·17-s − 0.235·18-s − 1.73·19-s − 0.138·20-s + 0.557·21-s − 0.172·22-s − 0.784·23-s − 0.204·24-s − 0.922·25-s + 0.0428·26-s + 0.192·27-s + 0.482·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6018 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6018 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.802715290\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.802715290\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 17 | \( 1 - T \) |
| 59 | \( 1 - T \) |
good | 5 | \( 1 + 0.621T + 5T^{2} \) |
| 7 | \( 1 - 2.55T + 7T^{2} \) |
| 11 | \( 1 - 0.810T + 11T^{2} \) |
| 13 | \( 1 + 0.218T + 13T^{2} \) |
| 19 | \( 1 + 7.58T + 19T^{2} \) |
| 23 | \( 1 + 3.76T + 23T^{2} \) |
| 29 | \( 1 - 6.91T + 29T^{2} \) |
| 31 | \( 1 + 1.43T + 31T^{2} \) |
| 37 | \( 1 + 2.07T + 37T^{2} \) |
| 41 | \( 1 - 5.09T + 41T^{2} \) |
| 43 | \( 1 - 10.7T + 43T^{2} \) |
| 47 | \( 1 + 2.11T + 47T^{2} \) |
| 53 | \( 1 - 6.69T + 53T^{2} \) |
| 61 | \( 1 + 2.33T + 61T^{2} \) |
| 67 | \( 1 - 11.9T + 67T^{2} \) |
| 71 | \( 1 - 14.6T + 71T^{2} \) |
| 73 | \( 1 - 6.78T + 73T^{2} \) |
| 79 | \( 1 - 14.4T + 79T^{2} \) |
| 83 | \( 1 - 5.08T + 83T^{2} \) |
| 89 | \( 1 - 18.4T + 89T^{2} \) |
| 97 | \( 1 + 9.44T + 97T^{2} \) |
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
−8.010613230104605754630165241904, −7.79945217505447874242679969413, −6.76761061732107275327045539035, −6.18526592500037716280226828841, −5.17140197269843749717428094103, −4.26195301396835247278301294550, −3.71245051086947769716863300357, −2.41593143244427406698702891629, −1.95275856808568802719210280750, −0.76340938214273792396636765566,
0.76340938214273792396636765566, 1.95275856808568802719210280750, 2.41593143244427406698702891629, 3.71245051086947769716863300357, 4.26195301396835247278301294550, 5.17140197269843749717428094103, 6.18526592500037716280226828841, 6.76761061732107275327045539035, 7.79945217505447874242679969413, 8.010613230104605754630165241904