L(s) = 1 | + (0.5 + 0.866i)2-s + (−0.5 − 0.866i)3-s + (−0.499 + 0.866i)4-s + (0.499 − 0.866i)6-s − 4·7-s − 0.999·8-s + (1 − 1.73i)9-s + 3·11-s + 0.999·12-s + (−1 + 1.73i)13-s + (−2 − 3.46i)14-s + (−0.5 − 0.866i)16-s + (3 + 5.19i)17-s + 2·18-s + (−3.5 + 2.59i)19-s + ⋯ |
L(s) = 1 | + (0.353 + 0.612i)2-s + (−0.288 − 0.499i)3-s + (−0.249 + 0.433i)4-s + (0.204 − 0.353i)6-s − 1.51·7-s − 0.353·8-s + (0.333 − 0.577i)9-s + 0.904·11-s + 0.288·12-s + (−0.277 + 0.480i)13-s + (−0.534 − 0.925i)14-s + (−0.125 − 0.216i)16-s + (0.727 + 1.26i)17-s + 0.471·18-s + (−0.802 + 0.596i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.910 - 0.412i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.910 - 0.412i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.751202 + 0.162355i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.751202 + 0.162355i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.5 - 0.866i)T \) |
| 19 | \( 1 + (3.5 - 2.59i)T \) |
good | 3 | \( 1 + (0.5 + 0.866i)T + (-1.5 + 2.59i)T^{2} \) |
| 5 | \( 1 + (-2.5 + 4.33i)T^{2} \) |
| 7 | \( 1 + 4T + 7T^{2} \) |
| 11 | \( 1 - 3T + 11T^{2} \) |
| 13 | \( 1 + (1 - 1.73i)T + (-6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 + (-3 - 5.19i)T + (-8.5 + 14.7i)T^{2} \) |
| 23 | \( 1 + (-3 + 5.19i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 - 2T + 31T^{2} \) |
| 37 | \( 1 + 10T + 37T^{2} \) |
| 41 | \( 1 + (4.5 + 7.79i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (-2 - 3.46i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (3 - 5.19i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (-4.5 - 7.79i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-2 + 3.46i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-3.5 + 6.06i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (-3 - 5.19i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (-0.5 - 0.866i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (-2 - 3.46i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 - 3T + 83T^{2} \) |
| 89 | \( 1 + (3 - 5.19i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (8.5 + 14.7i)T + (-48.5 + 84.0i)T^{2} \) |
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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
−16.56755263055327096788174289875, −15.28211244087574746901138481897, −14.16375082548199138628233952415, −12.63506419650880082403749061712, −12.32905909770147358575041978115, −10.15936809738882263373537801753, −8.779753188290020290978802133914, −6.83204172773892524372973604308, −6.22286096753511853920352949441, −3.82596749090966880616078458570,
3.35984729169594061369028138786, 5.16857224088052081785291055530, 6.90757797704302146249138961911, 9.299207338925210579842697562969, 10.12984738363830859482615609411, 11.45043803885142720835005436522, 12.73681987203851920762129508131, 13.66685992235296755885443165749, 15.20908919268970739743749330527, 16.22137496798641026167508889655