L(s) = 1 | + (0.959 + 1.03i)2-s + (2.75 − 1.58i)3-s + (−0.159 + 1.99i)4-s + (−0.5 + 0.866i)5-s + (4.29 + 1.33i)6-s + (−1.04 − 2.43i)7-s + (−2.22 + 1.74i)8-s + (3.55 − 6.15i)9-s + (−1.37 + 0.311i)10-s + (1.21 + 2.09i)11-s + (2.72 + 5.74i)12-s − 1.53·13-s + (1.52 − 3.41i)14-s + 3.17i·15-s + (−3.94 − 0.636i)16-s + (−6.58 + 3.79i)17-s + ⋯ |
L(s) = 1 | + (0.678 + 0.734i)2-s + (1.58 − 0.917i)3-s + (−0.0798 + 0.996i)4-s + (−0.223 + 0.387i)5-s + (1.75 + 0.545i)6-s + (−0.394 − 0.919i)7-s + (−0.786 + 0.617i)8-s + (1.18 − 2.05i)9-s + (−0.436 + 0.0984i)10-s + (0.364 + 0.631i)11-s + (0.788 + 1.65i)12-s − 0.426·13-s + (0.408 − 0.912i)14-s + 0.820i·15-s + (−0.987 − 0.159i)16-s + (−1.59 + 0.921i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.913 - 0.407i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.913 - 0.407i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.47626 + 0.528063i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.47626 + 0.528063i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.959 - 1.03i)T \) |
| 5 | \( 1 + (0.5 - 0.866i)T \) |
| 7 | \( 1 + (1.04 + 2.43i)T \) |
good | 3 | \( 1 + (-2.75 + 1.58i)T + (1.5 - 2.59i)T^{2} \) |
| 11 | \( 1 + (-1.21 - 2.09i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + 1.53T + 13T^{2} \) |
| 17 | \( 1 + (6.58 - 3.79i)T + (8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (1.52 + 0.883i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-5.66 - 3.26i)T + (11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + 2.34iT - 29T^{2} \) |
| 31 | \( 1 + (1.04 + 1.80i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (-2.47 - 1.42i)T + (18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + 6.90iT - 41T^{2} \) |
| 43 | \( 1 + 1.39T + 43T^{2} \) |
| 47 | \( 1 + (5.65 - 9.79i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (-6.82 + 3.93i)T + (26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (-4.90 + 2.82i)T + (29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-4.93 + 8.54i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-3.13 - 5.42i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + 3.98iT - 71T^{2} \) |
| 73 | \( 1 + (2.73 - 1.57i)T + (36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (1.75 + 1.01i)T + (39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 - 0.288iT - 83T^{2} \) |
| 89 | \( 1 + (-12.0 - 6.95i)T + (44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + 0.249iT - 97T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−12.52301893924254061414826676925, −11.19302490307914498207217005455, −9.636439157766313224816840817488, −8.735434649196209035314839618012, −7.78827477905641351027198960034, −7.00622267511669480616217682788, −6.54720085456883961619087292973, −4.36132908375098574048844558801, −3.50838850316983896707663663371, −2.26026799537562573251312934066,
2.32533198764623629518950548332, 3.14810677177489688169906533869, 4.30558179569597222966196803033, 5.17241762014810469023276979646, 6.81658786018162086084755751209, 8.610331097411360214213551277749, 8.960793796734820344041054159213, 9.764683514486455943047536768173, 10.83461025962533304362027281952, 11.80256924759210710269642329439