| L(s) = 1 | + (−1.81 − 1.46i)2-s + (0.238 + 0.971i)3-s + (0.729 + 3.38i)4-s + (−1.00 + 1.53i)5-s + (0.992 − 2.11i)6-s + (−4.01 + 1.04i)7-s + (1.53 − 3.04i)8-s + (−0.886 + 0.462i)9-s + (4.07 − 1.31i)10-s + (2.62 + 3.76i)11-s + (−3.11 + 1.51i)12-s + (−4.49 − 4.30i)13-s + (8.83 + 3.99i)14-s + (−1.72 − 0.610i)15-s + (−0.960 + 0.434i)16-s + (2.79 − 0.237i)17-s + ⋯ |
| L(s) = 1 | + (−1.28 − 1.03i)2-s + (0.137 + 0.560i)3-s + (0.364 + 1.69i)4-s + (−0.449 + 0.685i)5-s + (0.405 − 0.863i)6-s + (−1.51 + 0.395i)7-s + (0.543 − 1.07i)8-s + (−0.295 + 0.154i)9-s + (1.28 − 0.414i)10-s + (0.791 + 1.13i)11-s + (−0.898 + 0.437i)12-s + (−1.24 − 1.19i)13-s + (2.36 + 1.06i)14-s + (−0.446 − 0.157i)15-s + (−0.240 + 0.108i)16-s + (0.677 − 0.0576i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 669 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.828 + 0.560i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 669 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.828 + 0.560i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.0322167 - 0.105135i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0322167 - 0.105135i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-0.238 - 0.971i)T \) |
| 223 | \( 1 + (-7.17 + 13.0i)T \) |
| good | 2 | \( 1 + (1.81 + 1.46i)T + (0.421 + 1.95i)T^{2} \) |
| 5 | \( 1 + (1.00 - 1.53i)T + (-1.99 - 4.58i)T^{2} \) |
| 7 | \( 1 + (4.01 - 1.04i)T + (6.11 - 3.41i)T^{2} \) |
| 11 | \( 1 + (-2.62 - 3.76i)T + (-3.81 + 10.3i)T^{2} \) |
| 13 | \( 1 + (4.49 + 4.30i)T + (0.551 + 12.9i)T^{2} \) |
| 17 | \( 1 + (-2.79 + 0.237i)T + (16.7 - 2.87i)T^{2} \) |
| 19 | \( 1 + (2.81 - 1.68i)T + (9.03 - 16.7i)T^{2} \) |
| 23 | \( 1 + (3.27 + 0.185i)T + (22.8 + 2.59i)T^{2} \) |
| 29 | \( 1 + (-3.41 - 2.45i)T + (9.27 + 27.4i)T^{2} \) |
| 31 | \( 1 + (-6.96 + 8.38i)T + (-5.67 - 30.4i)T^{2} \) |
| 37 | \( 1 + (4.20 + 8.95i)T + (-23.6 + 28.4i)T^{2} \) |
| 41 | \( 1 + (3.71 + 9.24i)T + (-29.6 + 28.3i)T^{2} \) |
| 43 | \( 1 + (-0.772 + 0.0878i)T + (41.9 - 9.65i)T^{2} \) |
| 47 | \( 1 + (-0.721 - 0.430i)T + (22.3 + 41.3i)T^{2} \) |
| 53 | \( 1 + (-9.73 + 4.07i)T + (37.2 - 37.7i)T^{2} \) |
| 59 | \( 1 + (2.54 - 1.72i)T + (21.9 - 54.7i)T^{2} \) |
| 61 | \( 1 + (6.99 - 2.03i)T + (51.4 - 32.7i)T^{2} \) |
| 67 | \( 1 + (5.91 + 1.90i)T + (54.4 + 39.0i)T^{2} \) |
| 71 | \( 1 + (-0.222 - 3.14i)T + (-70.2 + 10.0i)T^{2} \) |
| 73 | \( 1 + (16.0 + 0.455i)T + (72.8 + 4.12i)T^{2} \) |
| 79 | \( 1 + (6.79 - 8.66i)T + (-18.8 - 76.7i)T^{2} \) |
| 83 | \( 1 + (-7.03 - 0.799i)T + (80.8 + 18.6i)T^{2} \) |
| 89 | \( 1 + (3.49 + 5.33i)T + (-35.5 + 81.6i)T^{2} \) |
| 97 | \( 1 + (-11.3 + 0.640i)T + (96.3 - 10.9i)T^{2} \) |
| 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
−10.22625044754160224462232112511, −9.581272120622939899053764555768, −8.835813934582242734948039136712, −7.69865629367721794628027926135, −7.02279647955758811816892233185, −5.71291460079227736734146792801, −4.01611643795932527652038366321, −3.10905782163524832296021882764, −2.33464281325195586665897654526, −0.10118733941508495472845618588,
1.11942076531717978145740515687, 3.13421434428486036100304410710, 4.56987122019830351605572587814, 6.21397315572292368641199845639, 6.54516428740903344287619683201, 7.35003550066745869447178988864, 8.420861256209297854227427315903, 8.834213260468326980434640691812, 9.752479112879149141346348627119, 10.31795447869841787146291932231
