| 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
−10.61731803255707141919051188579, −9.663146263014543987397305514360, −8.968087546051583194432931890998, −8.352322773310351476470853519832, −7.57159031002297062721691733612, −6.20948009409687592127225562837, −5.73550220670186939349904729299, −4.21043709353206773076948138447, −2.01289694052453147979383130861, −0.16057895545516383469301059701,
1.81824273636983133504785966789, 3.09956013946915902033109507893, 4.10828301994651221823502962139, 6.25511265304052321855805698158, 7.15390328451871765729485573339, 8.018765185094681621282799405051, 8.962481956267687294265329200994, 10.00522794377417160329755594833, 10.55326523914850883811324216845, 11.06598639092247768997876439105
