| L(s) = 1 | + (−1.25 + 0.655i)2-s + (−1.36 − 1.77i)3-s + (1.14 − 1.64i)4-s + (−1.70 + 2.22i)5-s + (2.87 + 1.33i)6-s + (−0.279 − 2.63i)7-s + (−0.355 + 2.80i)8-s + (−0.520 + 1.94i)9-s + (0.681 − 3.89i)10-s + (−0.192 + 1.46i)11-s + (−4.47 + 0.209i)12-s + (−0.943 + 2.27i)13-s + (2.07 + 3.11i)14-s + 6.26·15-s + (−1.39 − 3.74i)16-s + (−3.38 + 5.87i)17-s + ⋯ |
| L(s) = 1 | + (−0.886 + 0.463i)2-s + (−0.786 − 1.02i)3-s + (0.570 − 0.821i)4-s + (−0.762 + 0.993i)5-s + (1.17 + 0.544i)6-s + (−0.105 − 0.994i)7-s + (−0.125 + 0.992i)8-s + (−0.173 + 0.647i)9-s + (0.215 − 1.23i)10-s + (−0.0580 + 0.440i)11-s + (−1.29 + 0.0606i)12-s + (−0.261 + 0.631i)13-s + (0.554 + 0.832i)14-s + 1.61·15-s + (−0.348 − 0.937i)16-s + (−0.822 + 1.42i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 224 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.574 - 0.818i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 224 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.574 - 0.818i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.0985903 + 0.189537i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0985903 + 0.189537i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.25 - 0.655i)T \) |
| 7 | \( 1 + (0.279 + 2.63i)T \) |
| good | 3 | \( 1 + (1.36 + 1.77i)T + (-0.776 + 2.89i)T^{2} \) |
| 5 | \( 1 + (1.70 - 2.22i)T + (-1.29 - 4.82i)T^{2} \) |
| 11 | \( 1 + (0.192 - 1.46i)T + (-10.6 - 2.84i)T^{2} \) |
| 13 | \( 1 + (0.943 - 2.27i)T + (-9.19 - 9.19i)T^{2} \) |
| 17 | \( 1 + (3.38 - 5.87i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-1.08 - 8.24i)T + (-18.3 + 4.91i)T^{2} \) |
| 23 | \( 1 + (3.73 + 1.00i)T + (19.9 + 11.5i)T^{2} \) |
| 29 | \( 1 + (-1.45 + 3.50i)T + (-20.5 - 20.5i)T^{2} \) |
| 31 | \( 1 + (-2.28 + 3.96i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (1.86 + 1.43i)T + (9.57 + 35.7i)T^{2} \) |
| 41 | \( 1 + (2.36 - 2.36i)T - 41iT^{2} \) |
| 43 | \( 1 + (7.51 - 3.11i)T + (30.4 - 30.4i)T^{2} \) |
| 47 | \( 1 + (-6.98 + 4.03i)T + (23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (-0.948 - 0.124i)T + (51.1 + 13.7i)T^{2} \) |
| 59 | \( 1 + (-0.136 + 1.03i)T + (-56.9 - 15.2i)T^{2} \) |
| 61 | \( 1 + (-0.838 - 6.36i)T + (-58.9 + 15.7i)T^{2} \) |
| 67 | \( 1 + (-1.71 + 1.31i)T + (17.3 - 64.7i)T^{2} \) |
| 71 | \( 1 + (8.54 + 8.54i)T + 71iT^{2} \) |
| 73 | \( 1 + (2.13 + 7.94i)T + (-63.2 + 36.5i)T^{2} \) |
| 79 | \( 1 + (-4.04 - 7.00i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (14.6 + 6.05i)T + (58.6 + 58.6i)T^{2} \) |
| 89 | \( 1 + (3.44 - 12.8i)T + (-77.0 - 44.5i)T^{2} \) |
| 97 | \( 1 + 1.00iT - 97T^{2} \) |
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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
−12.23642468047709705623084617893, −11.54060944044882770822195859921, −10.61293036264379792465752157198, −9.952221170161821142266906188922, −8.160653481073856581920448482169, −7.48796248073426640881836823804, −6.69268114325437750806191840701, −6.05439860557288114847748644678, −4.02029071754326358269271562036, −1.75244255300036878618123818974,
0.25276808099701627853498579983, 2.89563641854230403556395459211, 4.52019148414474140432942872827, 5.36022362839987850185092953429, 7.00882283129488273023676757727, 8.431362179717926617456570945360, 9.029524991534478772576918383997, 9.930440340343782009709724964843, 11.08156883414935162270617499888, 11.66333388967083347692185171526
