L(s) = 1 | + 2.42·5-s + (0.710 − 2.54i)7-s − 4.70·11-s + (1.71 − 2.96i)13-s + (0.851 − 1.47i)17-s + (−0.641 − 1.11i)19-s + 1.12·23-s + 0.861·25-s + (−2.35 − 4.07i)29-s + (1.71 + 2.96i)31-s + (1.72 − 6.17i)35-s + (−4.27 − 7.40i)37-s + (1.85 − 3.21i)41-s + (−2.77 − 4.80i)43-s + (−5.91 + 10.2i)47-s + ⋯ |
L(s) = 1 | + 1.08·5-s + (0.268 − 0.963i)7-s − 1.41·11-s + (0.474 − 0.821i)13-s + (0.206 − 0.357i)17-s + (−0.147 − 0.254i)19-s + 0.234·23-s + 0.172·25-s + (−0.436 − 0.756i)29-s + (0.307 + 0.532i)31-s + (0.290 − 1.04i)35-s + (−0.702 − 1.21i)37-s + (0.290 − 0.502i)41-s + (−0.422 − 0.732i)43-s + (−0.862 + 1.49i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2268 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.160 + 0.987i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2268 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.160 + 0.987i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.731336236\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.731336236\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (-0.710 + 2.54i)T \) |
good | 5 | \( 1 - 2.42T + 5T^{2} \) |
| 11 | \( 1 + 4.70T + 11T^{2} \) |
| 13 | \( 1 + (-1.71 + 2.96i)T + (-6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 + (-0.851 + 1.47i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (0.641 + 1.11i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 - 1.12T + 23T^{2} \) |
| 29 | \( 1 + (2.35 + 4.07i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (-1.71 - 2.96i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (4.27 + 7.40i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (-1.85 + 3.21i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (2.77 + 4.80i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (5.91 - 10.2i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (-5.13 + 8.88i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (-2.06 - 3.57i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-4.62 + 8.01i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-5.56 - 9.63i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + 14.9T + 71T^{2} \) |
| 73 | \( 1 + (1.06 - 1.85i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-7.26 + 12.5i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (-4.21 - 7.29i)T + (-41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + (-8.04 - 13.9i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (8.12 + 14.0i)T + (-48.5 + 84.0i)T^{2} \) |
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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
−8.796692896736426266494557233050, −7.948288078507669519550112250639, −7.37045914313953812415309418917, −6.43530830998232782541566605038, −5.49971143905617084856615747622, −5.07749453900304426468190849621, −3.88213154673551061039267051827, −2.84942289485843443095687514348, −1.89318752447547470278514728944, −0.55934053428233781162252547679,
1.60328604644358116945927535363, 2.32838408177465230684447699241, 3.28651693260839478317117687738, 4.66551805215900006129743256043, 5.39674534490728219714360479834, 5.97789269348909436108653677855, 6.74828400661651925694410864469, 7.86098744761838504451925753545, 8.522553383391670258079733903695, 9.206489548721169758042805767932