| L(s) = 1 | + (1.40 − 0.113i)2-s + (1.57 − 1.79i)3-s + (1.97 − 0.319i)4-s + (−4.01 − 0.263i)5-s + (2.01 − 2.71i)6-s + (0.0172 − 2.64i)7-s + (2.74 − 0.674i)8-s + (−0.354 − 2.68i)9-s + (−5.69 + 0.0844i)10-s + (1.76 + 0.599i)11-s + (2.53 − 4.05i)12-s + (0.0225 + 0.0338i)13-s + (−0.275 − 3.73i)14-s + (−6.80 + 6.80i)15-s + (3.79 − 1.26i)16-s + (−6.25 + 1.67i)17-s + ⋯ |
| L(s) = 1 | + (0.996 − 0.0801i)2-s + (0.909 − 1.03i)3-s + (0.987 − 0.159i)4-s + (−1.79 − 0.117i)5-s + (0.823 − 1.10i)6-s + (0.00653 − 0.999i)7-s + (0.971 − 0.238i)8-s + (−0.118 − 0.896i)9-s + (−1.80 + 0.0267i)10-s + (0.532 + 0.180i)11-s + (0.732 − 1.16i)12-s + (0.00626 + 0.00937i)13-s + (−0.0736 − 0.997i)14-s + (−1.75 + 1.75i)15-s + (0.948 − 0.315i)16-s + (−1.51 + 0.406i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0517 + 0.998i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0517 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.92256 - 1.82546i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.92256 - 1.82546i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-1.40 + 0.113i)T \) |
| 7 | \( 1 + (-0.0172 + 2.64i)T \) |
| good | 3 | \( 1 + (-1.57 + 1.79i)T + (-0.391 - 2.97i)T^{2} \) |
| 5 | \( 1 + (4.01 + 0.263i)T + (4.95 + 0.652i)T^{2} \) |
| 11 | \( 1 + (-1.76 - 0.599i)T + (8.72 + 6.69i)T^{2} \) |
| 13 | \( 1 + (-0.0225 - 0.0338i)T + (-4.97 + 12.0i)T^{2} \) |
| 17 | \( 1 + (6.25 - 1.67i)T + (14.7 - 8.5i)T^{2} \) |
| 19 | \( 1 + (-5.65 - 2.79i)T + (11.5 + 15.0i)T^{2} \) |
| 23 | \( 1 + (-2.19 + 0.288i)T + (22.2 - 5.95i)T^{2} \) |
| 29 | \( 1 + (-1.07 + 5.40i)T + (-26.7 - 11.0i)T^{2} \) |
| 31 | \( 1 + (-0.0165 - 0.00957i)T + (15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (0.763 - 11.6i)T + (-36.6 - 4.82i)T^{2} \) |
| 41 | \( 1 + (0.510 - 1.23i)T + (-28.9 - 28.9i)T^{2} \) |
| 43 | \( 1 + (-0.363 + 0.0722i)T + (39.7 - 16.4i)T^{2} \) |
| 47 | \( 1 + (-0.928 + 3.46i)T + (-40.7 - 23.5i)T^{2} \) |
| 53 | \( 1 + (3.82 + 1.29i)T + (42.0 + 32.2i)T^{2} \) |
| 59 | \( 1 + (-0.486 - 0.986i)T + (-35.9 + 46.8i)T^{2} \) |
| 61 | \( 1 + (-3.71 - 10.9i)T + (-48.3 + 37.1i)T^{2} \) |
| 67 | \( 1 + (-5.42 + 6.18i)T + (-8.74 - 66.4i)T^{2} \) |
| 71 | \( 1 + (13.6 - 5.66i)T + (50.2 - 50.2i)T^{2} \) |
| 73 | \( 1 + (2.63 - 2.02i)T + (18.8 - 70.5i)T^{2} \) |
| 79 | \( 1 + (6.19 + 1.65i)T + (68.4 + 39.5i)T^{2} \) |
| 83 | \( 1 + (9.32 - 6.23i)T + (31.7 - 76.6i)T^{2} \) |
| 89 | \( 1 + (-7.34 + 9.57i)T + (-23.0 - 85.9i)T^{2} \) |
| 97 | \( 1 + 5.42iT - 97T^{2} \) |
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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
−11.44039579543502436967106650670, −10.23282929761514879608348763716, −8.616789102188705631874012270835, −7.82643494175192136392755105579, −7.21803245040479606454318128603, −6.57905946621126881642095786129, −4.64580759271802580638624380770, −3.89561541624884892728642390159, −2.97687855816036278570409523270, −1.28675699951865152571333052531,
2.73204717490498757234948028796, 3.48064074542132672877598624816, 4.33793690793738427459510894692, 5.13743162788484240997990904681, 6.74385237542536587759940127249, 7.62329665508961077853561963886, 8.696416662862339867402932771050, 9.223773685428088449668281697214, 10.81641671549133877412187344818, 11.39412228112428399022742990868
