L(s) = 1 | + (0.5 − 0.866i)2-s + (−0.499 − 0.866i)4-s + (2.34 + 1.35i)5-s + (0.222 + 2.63i)7-s − 0.999·8-s + (2.34 − 1.35i)10-s + (−2.77 + 1.82i)11-s + 2.18i·13-s + (2.39 + 1.12i)14-s + (−0.5 + 0.866i)16-s + (−0.0281 − 0.0488i)17-s + (−2.38 − 1.37i)19-s − 2.71i·20-s + (0.192 + 3.31i)22-s + (6.02 + 3.47i)23-s + ⋯ |
L(s) = 1 | + (0.353 − 0.612i)2-s + (−0.249 − 0.433i)4-s + (1.05 + 0.606i)5-s + (0.0839 + 0.996i)7-s − 0.353·8-s + (0.742 − 0.428i)10-s + (−0.835 + 0.549i)11-s + 0.604i·13-s + (0.639 + 0.300i)14-s + (−0.125 + 0.216i)16-s + (−0.00683 − 0.0118i)17-s + (−0.547 − 0.315i)19-s − 0.606i·20-s + (0.0409 + 0.705i)22-s + (1.25 + 0.725i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1386 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.681 - 0.731i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1386 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.681 - 0.731i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.000953406\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.000953406\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 7 | \( 1 + (-0.222 - 2.63i)T \) |
| 11 | \( 1 + (2.77 - 1.82i)T \) |
good | 5 | \( 1 + (-2.34 - 1.35i)T + (2.5 + 4.33i)T^{2} \) |
| 13 | \( 1 - 2.18iT - 13T^{2} \) |
| 17 | \( 1 + (0.0281 + 0.0488i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (2.38 + 1.37i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-6.02 - 3.47i)T + (11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + 6.43T + 29T^{2} \) |
| 31 | \( 1 + (-3.65 - 6.32i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (1.62 - 2.80i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + 2.71T + 41T^{2} \) |
| 43 | \( 1 - 10.6iT - 43T^{2} \) |
| 47 | \( 1 + (2.58 + 1.49i)T + (23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (-10.1 + 5.88i)T + (26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (-12.2 + 7.07i)T + (29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (10.2 + 5.93i)T + (30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-0.749 - 1.29i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + 1.82iT - 71T^{2} \) |
| 73 | \( 1 + (6.82 - 3.93i)T + (36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-7.82 - 4.51i)T + (39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 - 17.2T + 83T^{2} \) |
| 89 | \( 1 + (-7.43 - 4.29i)T + (44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 - 6.25T + 97T^{2} \) |
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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
−9.715315956060883865405369491918, −9.178606892915514965876850899297, −8.252274171641681555404103087988, −6.99608175451352125755315152305, −6.30146073533426477713026527826, −5.34461346877386254116475321571, −4.82086350942055966766903622468, −3.31682533681995778132166978092, −2.45020068582878693163722069238, −1.74952653981419131878330762358,
0.70286516382537098266320639000, 2.26726959993893001505409457043, 3.52760955473872882908135837397, 4.57458068849963658432923420005, 5.42608860631479057111136292839, 5.97223452124013170903440482875, 7.02965958073164858728929847588, 7.76905548156897837065521108961, 8.617801144791402146692332988742, 9.319040828352880836205963593901