| L(s) = 1 | − 3.92·2-s − 3·3-s + 7.39·4-s − 14.7·5-s + 11.7·6-s − 6.54·7-s + 2.37·8-s + 9·9-s + 58.0·10-s + 46.3·11-s − 22.1·12-s + 23.3·13-s + 25.6·14-s + 44.3·15-s − 68.4·16-s + 85.8·17-s − 35.3·18-s + 117.·19-s − 109.·20-s + 19.6·21-s − 181.·22-s + 129.·23-s − 7.12·24-s + 93.6·25-s − 91.4·26-s − 27·27-s − 48.4·28-s + ⋯ |
| L(s) = 1 | − 1.38·2-s − 0.577·3-s + 0.924·4-s − 1.32·5-s + 0.800·6-s − 0.353·7-s + 0.104·8-s + 0.333·9-s + 1.83·10-s + 1.27·11-s − 0.533·12-s + 0.497·13-s + 0.490·14-s + 0.763·15-s − 1.06·16-s + 1.22·17-s − 0.462·18-s + 1.41·19-s − 1.22·20-s + 0.204·21-s − 1.76·22-s + 1.16·23-s − 0.0605·24-s + 0.749·25-s − 0.690·26-s − 0.192·27-s − 0.326·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1011 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1011 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.6550188015\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6550188015\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + 3T \) |
| 337 | \( 1 + 337T \) |
| good | 2 | \( 1 + 3.92T + 8T^{2} \) |
| 5 | \( 1 + 14.7T + 125T^{2} \) |
| 7 | \( 1 + 6.54T + 343T^{2} \) |
| 11 | \( 1 - 46.3T + 1.33e3T^{2} \) |
| 13 | \( 1 - 23.3T + 2.19e3T^{2} \) |
| 17 | \( 1 - 85.8T + 4.91e3T^{2} \) |
| 19 | \( 1 - 117.T + 6.85e3T^{2} \) |
| 23 | \( 1 - 129.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 230.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 54.5T + 2.97e4T^{2} \) |
| 37 | \( 1 - 43.0T + 5.06e4T^{2} \) |
| 41 | \( 1 + 198.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 214.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 547.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 279.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 678.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 163.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 780.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 643.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 566.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 76.9T + 4.93e5T^{2} \) |
| 83 | \( 1 - 725.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 984.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 760.T + 9.12e5T^{2} \) |
| 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
−9.408873700480700858363028977855, −8.933655728330018658465269787591, −7.79290093282959412231875678251, −7.40701057575083344605096433410, −6.55441888348865193674155558607, −5.35319560175500931376712931038, −4.10809286363244382880727877339, −3.31162376260262803661968671374, −1.32764627902827761150029119957, −0.64036560285054454949872817576,
0.64036560285054454949872817576, 1.32764627902827761150029119957, 3.31162376260262803661968671374, 4.10809286363244382880727877339, 5.35319560175500931376712931038, 6.55441888348865193674155558607, 7.40701057575083344605096433410, 7.79290093282959412231875678251, 8.933655728330018658465269787591, 9.408873700480700858363028977855
