| L(s) = 1 | + (0.433 + 2.12i)2-s + (−0.987 − 0.160i)3-s + (−2.48 + 1.05i)4-s + (2.08 − 0.512i)5-s + (−0.0873 − 2.16i)6-s + (0.507 − 1.07i)7-s + (−0.867 − 1.25i)8-s + (0.948 + 0.316i)9-s + (1.99 + 4.19i)10-s + (3.00 − 1.00i)11-s + (2.62 − 0.647i)12-s + (2.78 + 2.28i)13-s + (2.49 + 0.614i)14-s + (−2.13 + 0.172i)15-s + (−1.45 + 1.51i)16-s + (0.867 − 1.82i)17-s + ⋯ |
| L(s) = 1 | + (0.306 + 1.50i)2-s + (−0.569 − 0.0926i)3-s + (−1.24 + 0.529i)4-s + (0.930 − 0.229i)5-s + (−0.0356 − 0.884i)6-s + (0.191 − 0.404i)7-s + (−0.306 − 0.444i)8-s + (0.316 + 0.105i)9-s + (0.630 + 1.32i)10-s + (0.907 − 0.302i)11-s + (0.757 − 0.186i)12-s + (0.773 + 0.634i)13-s + (0.666 + 0.164i)14-s + (−0.551 + 0.0445i)15-s + (−0.363 + 0.377i)16-s + (0.210 − 0.443i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.315 - 0.948i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.315 - 0.948i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.00765 + 1.39671i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.00765 + 1.39671i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (0.987 + 0.160i)T \) |
| 13 | \( 1 + (-2.78 - 2.28i)T \) |
| good | 2 | \( 1 + (-0.433 - 2.12i)T + (-1.83 + 0.783i)T^{2} \) |
| 5 | \( 1 + (-2.08 + 0.512i)T + (4.42 - 2.32i)T^{2} \) |
| 7 | \( 1 + (-0.507 + 1.07i)T + (-4.42 - 5.42i)T^{2} \) |
| 11 | \( 1 + (-3.00 + 1.00i)T + (8.79 - 6.60i)T^{2} \) |
| 17 | \( 1 + (-0.867 + 1.82i)T + (-10.7 - 13.1i)T^{2} \) |
| 19 | \( 1 + (0.0764 + 0.132i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (1.47 - 2.54i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (-0.552 - 2.70i)T + (-26.6 + 11.3i)T^{2} \) |
| 31 | \( 1 + (-5.40 - 2.83i)T + (17.6 + 25.5i)T^{2} \) |
| 37 | \( 1 + (-1.97 + 1.25i)T + (15.8 - 33.4i)T^{2} \) |
| 41 | \( 1 + (9.27 + 1.50i)T + (38.8 + 12.9i)T^{2} \) |
| 43 | \( 1 + (3.47 + 2.19i)T + (18.4 + 38.8i)T^{2} \) |
| 47 | \( 1 + (1.11 + 9.21i)T + (-45.6 + 11.2i)T^{2} \) |
| 53 | \( 1 + (-4.39 - 6.36i)T + (-18.7 + 49.5i)T^{2} \) |
| 59 | \( 1 + (6.50 + 6.77i)T + (-2.37 + 58.9i)T^{2} \) |
| 61 | \( 1 + (-7.61 - 0.614i)T + (60.2 + 9.78i)T^{2} \) |
| 67 | \( 1 + (4.23 + 1.80i)T + (46.4 + 48.3i)T^{2} \) |
| 71 | \( 1 + (2.48 - 3.04i)T + (-14.2 - 69.5i)T^{2} \) |
| 73 | \( 1 + (-6.58 - 5.83i)T + (8.79 + 72.4i)T^{2} \) |
| 79 | \( 1 + (0.789 + 6.50i)T + (-76.7 + 18.9i)T^{2} \) |
| 83 | \( 1 + (2.51 - 6.64i)T + (-62.1 - 55.0i)T^{2} \) |
| 89 | \( 1 + (-4.83 + 8.36i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (0.282 + 0.975i)T + (-81.9 + 51.8i)T^{2} \) |
| 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
−11.24769867939277661654640141542, −10.14547114155465933552812254323, −9.139093311190946408603132939165, −8.372763663888194812955499423206, −7.16570160521405788391651375105, −6.51712412953259831677340688116, −5.78153049436612628326570390770, −4.96250202346834800970674213549, −3.86994181593379819372774033635, −1.54429831300458787475201697825,
1.26090566284036113316779662030, 2.36085477518004863972276028835, 3.64741024266005054790482019589, 4.69484192998365115140506049368, 5.85038492559729852199462614005, 6.59124861522503132876203575881, 8.240636814241605692579955173538, 9.383124157276233373180201565863, 10.08558456739218127399734430746, 10.63594268548350135712419091490
