| L(s) = 1 | + (−0.746 + 2.29i)2-s + (−1.61 + 1.79i)3-s + (−3.09 − 2.25i)4-s + (−0.5 + 0.866i)5-s + (−2.91 − 5.04i)6-s + (−0.252 + 2.40i)7-s + (3.57 − 2.59i)8-s + (−0.295 − 2.81i)9-s + (−1.61 − 1.79i)10-s + (−4.78 + 2.13i)11-s + (9.04 − 1.92i)12-s + (−1.78 − 0.380i)13-s + (−5.32 − 2.37i)14-s + (−0.746 − 2.29i)15-s + (0.927 + 2.85i)16-s + (−0.156 − 0.0697i)17-s + ⋯ |
| L(s) = 1 | + (−0.527 + 1.62i)2-s + (−0.932 + 1.03i)3-s + (−1.54 − 1.12i)4-s + (−0.223 + 0.387i)5-s + (−1.18 − 2.06i)6-s + (−0.0953 + 0.907i)7-s + (1.26 − 0.917i)8-s + (−0.0985 − 0.937i)9-s + (−0.510 − 0.567i)10-s + (−1.44 + 0.642i)11-s + (2.60 − 0.554i)12-s + (−0.496 − 0.105i)13-s + (−1.42 − 0.633i)14-s + (−0.192 − 0.592i)15-s + (0.231 + 0.713i)16-s + (−0.0380 − 0.0169i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 961 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.907 - 0.420i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 961 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.907 - 0.420i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.132982 + 0.0293475i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.132982 + 0.0293475i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 31 | \( 1 \) |
| good | 2 | \( 1 + (0.746 - 2.29i)T + (-1.61 - 1.17i)T^{2} \) |
| 3 | \( 1 + (1.61 - 1.79i)T + (-0.313 - 2.98i)T^{2} \) |
| 5 | \( 1 + (0.5 - 0.866i)T + (-2.5 - 4.33i)T^{2} \) |
| 7 | \( 1 + (0.252 - 2.40i)T + (-6.84 - 1.45i)T^{2} \) |
| 11 | \( 1 + (4.78 - 2.13i)T + (7.36 - 8.17i)T^{2} \) |
| 13 | \( 1 + (1.78 + 0.380i)T + (11.8 + 5.28i)T^{2} \) |
| 17 | \( 1 + (0.156 + 0.0697i)T + (11.3 + 12.6i)T^{2} \) |
| 19 | \( 1 + (1.55 - 0.329i)T + (17.3 - 7.72i)T^{2} \) |
| 23 | \( 1 + (-3.23 + 2.35i)T + (7.10 - 21.8i)T^{2} \) |
| 29 | \( 1 + (0.362 - 1.11i)T + (-23.4 - 17.0i)T^{2} \) |
| 37 | \( 1 + (0.5 + 0.866i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (-6.34 - 7.04i)T + (-4.28 + 40.7i)T^{2} \) |
| 43 | \( 1 + (8.70 - 1.85i)T + (39.2 - 17.4i)T^{2} \) |
| 47 | \( 1 + (0.511 + 1.57i)T + (-38.0 + 27.6i)T^{2} \) |
| 53 | \( 1 + (0.0179 + 0.170i)T + (-51.8 + 11.0i)T^{2} \) |
| 59 | \( 1 + (6.73 - 7.48i)T + (-6.16 - 58.6i)T^{2} \) |
| 61 | \( 1 - 2.82T + 61T^{2} \) |
| 67 | \( 1 + (-2.62 + 4.54i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (-1.47 - 13.9i)T + (-69.4 + 14.7i)T^{2} \) |
| 73 | \( 1 + (-3.49 + 1.55i)T + (48.8 - 54.2i)T^{2} \) |
| 79 | \( 1 + (13.9 + 6.19i)T + (52.8 + 58.7i)T^{2} \) |
| 83 | \( 1 + (-2.72 - 3.02i)T + (-8.67 + 82.5i)T^{2} \) |
| 89 | \( 1 + (-10.1 - 7.33i)T + (27.5 + 84.6i)T^{2} \) |
| 97 | \( 1 + (8.76 + 6.36i)T + (29.9 + 92.2i)T^{2} \) |
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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
−10.59032877182178843261874402157, −9.928357690541165409134411896900, −9.228720792761935524162695705653, −8.306143567862991911076112026459, −7.48385237612795340452487908295, −6.65311638790666477974662739897, −5.68845773958275171295976654104, −5.16601119057343687451344512418, −4.50992295036240723339751167371, −2.76280436984695769166294221627,
0.11697008111547050225485446860, 0.871383615329316795137008687303, 2.13986263610596689130360466315, 3.28966278909694531761564334211, 4.50881796605162494804224501004, 5.47719619875758954572112821991, 6.72090019180504523587761992351, 7.64228958990930700320042762089, 8.336233344637807299890464314645, 9.375240189631229967738591691188
