| L(s) = 1 | − 2-s − 3-s + 4-s − 2.92·5-s + 6-s − 1.10·7-s − 8-s + 9-s + 2.92·10-s + 11-s − 12-s − 6.35·13-s + 1.10·14-s + 2.92·15-s + 16-s + 5.76·17-s − 18-s + 5.38·19-s − 2.92·20-s + 1.10·21-s − 22-s − 7.93·23-s + 24-s + 3.53·25-s + 6.35·26-s − 27-s − 1.10·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 0.5·4-s − 1.30·5-s + 0.408·6-s − 0.416·7-s − 0.353·8-s + 0.333·9-s + 0.923·10-s + 0.301·11-s − 0.288·12-s − 1.76·13-s + 0.294·14-s + 0.754·15-s + 0.250·16-s + 1.39·17-s − 0.235·18-s + 1.23·19-s − 0.653·20-s + 0.240·21-s − 0.213·22-s − 1.65·23-s + 0.204·24-s + 0.706·25-s + 1.24·26-s − 0.192·27-s − 0.208·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4026 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4026 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.3410478501\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3410478501\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 11 | \( 1 - T \) |
| 61 | \( 1 + T \) |
| good | 5 | \( 1 + 2.92T + 5T^{2} \) |
| 7 | \( 1 + 1.10T + 7T^{2} \) |
| 13 | \( 1 + 6.35T + 13T^{2} \) |
| 17 | \( 1 - 5.76T + 17T^{2} \) |
| 19 | \( 1 - 5.38T + 19T^{2} \) |
| 23 | \( 1 + 7.93T + 23T^{2} \) |
| 29 | \( 1 - 3.35T + 29T^{2} \) |
| 31 | \( 1 + 4.75T + 31T^{2} \) |
| 37 | \( 1 + 10.4T + 37T^{2} \) |
| 41 | \( 1 - 4.86T + 41T^{2} \) |
| 43 | \( 1 + 5.07T + 43T^{2} \) |
| 47 | \( 1 + 6.64T + 47T^{2} \) |
| 53 | \( 1 + 4.48T + 53T^{2} \) |
| 59 | \( 1 + 0.0442T + 59T^{2} \) |
| 67 | \( 1 - 2.22T + 67T^{2} \) |
| 71 | \( 1 + 5.48T + 71T^{2} \) |
| 73 | \( 1 + 12.2T + 73T^{2} \) |
| 79 | \( 1 - 10.3T + 79T^{2} \) |
| 83 | \( 1 - 9.19T + 83T^{2} \) |
| 89 | \( 1 + 5.36T + 89T^{2} \) |
| 97 | \( 1 + 12.0T + 97T^{2} \) |
| 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
−8.231446756737157004657185824772, −7.59263158573263444660890699990, −7.31309094756982507359614613098, −6.42008080816589872744348602920, −5.46918863810733847452111015865, −4.76136602370692163110558971275, −3.69281093455108603487321660543, −3.07375378136506432264168546872, −1.70144266318788052513335336424, −0.37774563507719881413310557748,
0.37774563507719881413310557748, 1.70144266318788052513335336424, 3.07375378136506432264168546872, 3.69281093455108603487321660543, 4.76136602370692163110558971275, 5.46918863810733847452111015865, 6.42008080816589872744348602920, 7.31309094756982507359614613098, 7.59263158573263444660890699990, 8.231446756737157004657185824772
