| L(s) = 1 | + 2.90·5-s + i·7-s + 1.28i·11-s − 1.86i·13-s − 3.87i·17-s + 2.04·19-s + 0.934·23-s + 3.45·25-s + 2.98·29-s − 1.85i·31-s + 2.90i·35-s + 3.85i·37-s + 8.72i·41-s + 1.19·43-s + 11.1·47-s + ⋯ |
| L(s) = 1 | + 1.30·5-s + 0.377i·7-s + 0.388i·11-s − 0.518i·13-s − 0.939i·17-s + 0.468·19-s + 0.194·23-s + 0.691·25-s + 0.553·29-s − 0.333i·31-s + 0.491i·35-s + 0.634i·37-s + 1.36i·41-s + 0.181·43-s + 1.62·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.997 + 0.0726i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.997 + 0.0726i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.743138743\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.743138743\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 - iT \) |
| good | 5 | \( 1 - 2.90T + 5T^{2} \) |
| 11 | \( 1 - 1.28iT - 11T^{2} \) |
| 13 | \( 1 + 1.86iT - 13T^{2} \) |
| 17 | \( 1 + 3.87iT - 17T^{2} \) |
| 19 | \( 1 - 2.04T + 19T^{2} \) |
| 23 | \( 1 - 0.934T + 23T^{2} \) |
| 29 | \( 1 - 2.98T + 29T^{2} \) |
| 31 | \( 1 + 1.85iT - 31T^{2} \) |
| 37 | \( 1 - 3.85iT - 37T^{2} \) |
| 41 | \( 1 - 8.72iT - 41T^{2} \) |
| 43 | \( 1 - 1.19T + 43T^{2} \) |
| 47 | \( 1 - 11.1T + 47T^{2} \) |
| 53 | \( 1 + 4.43T + 53T^{2} \) |
| 59 | \( 1 + 10.9iT - 59T^{2} \) |
| 61 | \( 1 + 4.70iT - 61T^{2} \) |
| 67 | \( 1 + 4.41T + 67T^{2} \) |
| 71 | \( 1 - 9.72T + 71T^{2} \) |
| 73 | \( 1 - 9.40T + 73T^{2} \) |
| 79 | \( 1 + 16.4iT - 79T^{2} \) |
| 83 | \( 1 + 3.72iT - 83T^{2} \) |
| 89 | \( 1 + 12.2iT - 89T^{2} \) |
| 97 | \( 1 - 15.8T + 97T^{2} \) |
| 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
−7.999993805666755006730945091222, −7.39394982464681380197232107619, −6.42588007776293202188330407750, −6.04655500021998050743474809982, −5.10387293716500272268467175088, −4.79142468368845193630252276731, −3.43452334477851557937140476042, −2.65067035044115235795963660064, −1.94562784674886938832036061289, −0.855583516188966316940702149403,
0.941028894050565138960152363308, 1.86405854901708987435443960316, 2.64304732374113656447377092895, 3.69048329702704084171374355844, 4.41995069474313439754702016037, 5.50324070671641611127100146932, 5.79683596507269933627650659445, 6.66998450663107373880171848041, 7.21361850389308723864662966502, 8.168524578777860747595172646076
