L(s) = 1 | + (−0.5 − 0.866i)2-s + (−0.499 + 0.866i)4-s − 4·7-s + 0.999·8-s − 3·11-s + (−1 + 1.73i)13-s + (2 + 3.46i)14-s + (−0.5 − 0.866i)16-s + (−3 − 5.19i)17-s + (−3.5 + 2.59i)19-s + (1.5 + 2.59i)22-s + (−3 + 5.19i)23-s + (2.5 − 4.33i)25-s + 1.99·26-s + (1.99 − 3.46i)28-s + ⋯ |
L(s) = 1 | + (−0.353 − 0.612i)2-s + (−0.249 + 0.433i)4-s − 1.51·7-s + 0.353·8-s − 0.904·11-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 + (−0.802 + 0.596i)19-s + (0.319 + 0.553i)22-s + (−0.625 + 1.08i)23-s + (0.5 − 0.866i)25-s + 0.392·26-s + (0.377 − 0.654i)28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 342 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.813 - 0.582i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 342 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.813 - 0.582i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.5 + 0.866i)T \) |
| 3 | \( 1 \) |
| 19 | \( 1 + (3.5 - 2.59i)T \) |
good | 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
−10.86359945672895912011594580634, −9.942099990391009457814331411214, −9.416912148206655431557957120388, −8.318892326007540486521642731569, −7.16149101891613482558801324022, −6.22064754116218544873651323276, −4.77064755288893600186099684662, −3.42891521678550016338868234598, −2.36476060363458436705570941796, 0,
2.57383207088599541771277817450, 4.04652823208039659369000603373, 5.48297144535269949111404744615, 6.41980706726341948955153101582, 7.19989160102846856801864937797, 8.434921835077389512567977846217, 9.130308608174935567729676313094, 10.40600411846785924267505787741, 10.57709750828962052632580498825