| L(s) = 1 | + (−1.94 − 1.36i)2-s + (1.25 + 3.43i)4-s + (−0.278 + 2.21i)5-s + (−1.02 − 2.19i)7-s + (1.02 − 3.81i)8-s + (3.57 − 3.94i)10-s + (−0.00275 − 0.00327i)11-s + (0.792 + 1.13i)13-s + (−1.00 + 5.68i)14-s + (−1.59 + 1.33i)16-s + (−0.510 − 1.90i)17-s + (−6.69 + 3.86i)19-s + (−7.98 + 1.81i)20-s + (0.000887 + 0.0101i)22-s + (−6.70 − 3.12i)23-s + ⋯ |
| L(s) = 1 | + (−1.37 − 0.964i)2-s + (0.626 + 1.71i)4-s + (−0.124 + 0.992i)5-s + (−0.387 − 0.830i)7-s + (0.361 − 1.34i)8-s + (1.12 − 1.24i)10-s + (−0.000829 − 0.000988i)11-s + (0.219 + 0.313i)13-s + (−0.267 + 1.51i)14-s + (−0.398 + 0.334i)16-s + (−0.123 − 0.461i)17-s + (−1.53 + 0.886i)19-s + (−1.78 + 0.406i)20-s + (0.000189 + 0.00216i)22-s + (−1.39 − 0.651i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.318 - 0.948i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.318 - 0.948i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.0966400 + 0.134368i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0966400 + 0.134368i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 + (0.278 - 2.21i)T \) |
| good | 2 | \( 1 + (1.94 + 1.36i)T + (0.684 + 1.87i)T^{2} \) |
| 7 | \( 1 + (1.02 + 2.19i)T + (-4.49 + 5.36i)T^{2} \) |
| 11 | \( 1 + (0.00275 + 0.00327i)T + (-1.91 + 10.8i)T^{2} \) |
| 13 | \( 1 + (-0.792 - 1.13i)T + (-4.44 + 12.2i)T^{2} \) |
| 17 | \( 1 + (0.510 + 1.90i)T + (-14.7 + 8.5i)T^{2} \) |
| 19 | \( 1 + (6.69 - 3.86i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (6.70 + 3.12i)T + (14.7 + 17.6i)T^{2} \) |
| 29 | \( 1 + (-1.74 - 9.87i)T + (-27.2 + 9.91i)T^{2} \) |
| 31 | \( 1 + (5.62 - 2.04i)T + (23.7 - 19.9i)T^{2} \) |
| 37 | \( 1 + (-1.68 + 0.451i)T + (32.0 - 18.5i)T^{2} \) |
| 41 | \( 1 + (-2.95 - 0.520i)T + (38.5 + 14.0i)T^{2} \) |
| 43 | \( 1 + (0.628 - 7.18i)T + (-42.3 - 7.46i)T^{2} \) |
| 47 | \( 1 + (1.86 - 0.871i)T + (30.2 - 36.0i)T^{2} \) |
| 53 | \( 1 + (1.25 + 1.25i)T + 53iT^{2} \) |
| 59 | \( 1 + (0.763 + 0.640i)T + (10.2 + 58.1i)T^{2} \) |
| 61 | \( 1 + (6.39 + 2.32i)T + (46.7 + 39.2i)T^{2} \) |
| 67 | \( 1 + (7.45 - 5.21i)T + (22.9 - 62.9i)T^{2} \) |
| 71 | \( 1 + (8.32 + 4.80i)T + (35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (2.77 + 0.744i)T + (63.2 + 36.5i)T^{2} \) |
| 79 | \( 1 + (-6.95 + 1.22i)T + (74.2 - 27.0i)T^{2} \) |
| 83 | \( 1 + (5.84 - 8.34i)T + (-28.3 - 77.9i)T^{2} \) |
| 89 | \( 1 + (3.03 + 5.26i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (-12.5 - 1.09i)T + (95.5 + 16.8i)T^{2} \) |
| 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
−11.06598639092247768997876439105, −10.55326523914850883811324216845, −10.00522794377417160329755594833, −8.962481956267687294265329200994, −8.018765185094681621282799405051, −7.15390328451871765729485573339, −6.25511265304052321855805698158, −4.10828301994651221823502962139, −3.09956013946915902033109507893, −1.81824273636983133504785966789,
0.16057895545516383469301059701, 2.01289694052453147979383130861, 4.21043709353206773076948138447, 5.73550220670186939349904729299, 6.20948009409687592127225562837, 7.57159031002297062721691733612, 8.352322773310351476470853519832, 8.968087546051583194432931890998, 9.663146263014543987397305514360, 10.61731803255707141919051188579
