| L(s) = 1 | − 2-s − 3-s + 4-s − 0.582·5-s + 6-s + 1.07·7-s − 8-s + 9-s + 0.582·10-s − 11-s − 12-s − 7.09·13-s − 1.07·14-s + 0.582·15-s + 16-s − 4.95·17-s − 18-s + 1.60·19-s − 0.582·20-s − 1.07·21-s + 22-s − 2.09·23-s + 24-s − 4.66·25-s + 7.09·26-s − 27-s + 1.07·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 0.5·4-s − 0.260·5-s + 0.408·6-s + 0.407·7-s − 0.353·8-s + 0.333·9-s + 0.184·10-s − 0.301·11-s − 0.288·12-s − 1.96·13-s − 0.288·14-s + 0.150·15-s + 0.250·16-s − 1.20·17-s − 0.235·18-s + 0.368·19-s − 0.130·20-s − 0.235·21-s + 0.213·22-s − 0.436·23-s + 0.204·24-s − 0.932·25-s + 1.39·26-s − 0.192·27-s + 0.203·28-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
| \(L(1)\) |
\(\approx\) |
\(0.5520660159\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5520660159\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 0.582T + 5T^{2} \) |
| 7 | \( 1 - 1.07T + 7T^{2} \) |
| 13 | \( 1 + 7.09T + 13T^{2} \) |
| 17 | \( 1 + 4.95T + 17T^{2} \) |
| 19 | \( 1 - 1.60T + 19T^{2} \) |
| 23 | \( 1 + 2.09T + 23T^{2} \) |
| 29 | \( 1 - 5.03T + 29T^{2} \) |
| 31 | \( 1 + 6.40T + 31T^{2} \) |
| 37 | \( 1 - 3.98T + 37T^{2} \) |
| 41 | \( 1 - 5.98T + 41T^{2} \) |
| 43 | \( 1 + 9.28T + 43T^{2} \) |
| 47 | \( 1 - 5.53T + 47T^{2} \) |
| 53 | \( 1 - 12.2T + 53T^{2} \) |
| 59 | \( 1 - 0.912T + 59T^{2} \) |
| 67 | \( 1 + 2.78T + 67T^{2} \) |
| 71 | \( 1 - 0.183T + 71T^{2} \) |
| 73 | \( 1 - 9.52T + 73T^{2} \) |
| 79 | \( 1 - 8.83T + 79T^{2} \) |
| 83 | \( 1 - 7.97T + 83T^{2} \) |
| 89 | \( 1 + 3.41T + 89T^{2} \) |
| 97 | \( 1 - 7.95T + 97T^{2} \) |
| 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
−8.333770814847691600650029745668, −7.67317131256650199065816125042, −7.14708199670835753208437537182, −6.41040385046658115039193811126, −5.42433955039030729084351713617, −4.81090296525910462922049366642, −3.95968686485198863054977055919, −2.60861445074669587574554746088, −1.94257861139528914621829763754, −0.46416496969340687775913058451,
0.46416496969340687775913058451, 1.94257861139528914621829763754, 2.60861445074669587574554746088, 3.95968686485198863054977055919, 4.81090296525910462922049366642, 5.42433955039030729084351713617, 6.41040385046658115039193811126, 7.14708199670835753208437537182, 7.67317131256650199065816125042, 8.333770814847691600650029745668
