L(s) = 1 | + (0.5 − 0.866i)2-s + (0.130 + 1.72i)3-s + (−0.499 − 0.866i)4-s + (0.866 + 0.5i)5-s + (1.56 + 0.750i)6-s − 4.16·7-s − 0.999·8-s + (−2.96 + 0.449i)9-s + (0.866 − 0.499i)10-s + 2.96i·11-s + (1.43 − 0.976i)12-s + (−5.88 + 3.39i)13-s + (−2.08 + 3.60i)14-s + (−0.750 + 1.56i)15-s + (−0.5 + 0.866i)16-s + (1.56 + 0.900i)17-s + ⋯ |
L(s) = 1 | + (0.353 − 0.612i)2-s + (0.0750 + 0.997i)3-s + (−0.249 − 0.433i)4-s + (0.387 + 0.223i)5-s + (0.637 + 0.306i)6-s − 1.57·7-s − 0.353·8-s + (−0.988 + 0.149i)9-s + (0.273 − 0.158i)10-s + 0.894i·11-s + (0.413 − 0.281i)12-s + (−1.63 + 0.942i)13-s + (−0.556 + 0.963i)14-s + (−0.193 + 0.402i)15-s + (−0.125 + 0.216i)16-s + (0.378 + 0.218i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.580 - 0.813i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.580 - 0.813i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.363895 + 0.706780i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.363895 + 0.706780i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.5 + 0.866i)T \) |
| 3 | \( 1 + (-0.130 - 1.72i)T \) |
| 5 | \( 1 + (-0.866 - 0.5i)T \) |
| 19 | \( 1 + (-0.334 + 4.34i)T \) |
good | 7 | \( 1 + 4.16T + 7T^{2} \) |
| 11 | \( 1 - 2.96iT - 11T^{2} \) |
| 13 | \( 1 + (5.88 - 3.39i)T + (6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 + (-1.56 - 0.900i)T + (8.5 + 14.7i)T^{2} \) |
| 23 | \( 1 + (0.309 - 0.178i)T + (11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (-3.63 - 6.30i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 - 3.56iT - 31T^{2} \) |
| 37 | \( 1 + 6.14iT - 37T^{2} \) |
| 41 | \( 1 + (-3.81 + 6.60i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (5.13 - 8.89i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (4.46 - 2.57i)T + (23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (-1.75 - 3.03i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (4.38 - 7.59i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-1.88 - 3.27i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-0.922 + 0.532i)T + (33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (2.05 - 3.56i)T + (-35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + (-6.64 + 11.5i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-1.10 - 0.638i)T + (39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + 1.93iT - 83T^{2} \) |
| 89 | \( 1 + (-2.94 - 5.10i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (-11.2 - 6.49i)T + (48.5 + 84.0i)T^{2} \) |
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
−10.79408123437726398909080833134, −10.10060494623284546817510125848, −9.481425832891661471571801509934, −9.077225495854165625222840312813, −7.24308576939137986589380839977, −6.42143516249119085878484775200, −5.19931198711585633907208792673, −4.40548752239510075286814911839, −3.21639535464609906837697506586, −2.40699172895905961718422223384,
0.36738567265600272131819223193, 2.59910262143623417353885533101, 3.43968516099258092810098337797, 5.21147082320561708817812496343, 6.03573041622717882760490961277, 6.64227080404426614652912823678, 7.68849870413735379842424487361, 8.353256606944114957589484954412, 9.596596602210052720997861511829, 10.10507582544600379544493705940