L(s) = 1 | + (−1.25 + 2.17i)2-s + (−2.14 − 3.71i)4-s + (−0.5 + 0.866i)5-s − 0.221·7-s + 5.72·8-s + (−1.25 − 2.17i)10-s + 0.778·11-s + (2.5 + 4.33i)13-s + (0.278 − 0.481i)14-s + (−2.89 + 5.01i)16-s + (3.53 − 6.12i)17-s + (1.33 − 4.15i)19-s + 4.28·20-s + (−0.975 + 1.68i)22-s + (4.03 + 6.99i)23-s + ⋯ |
L(s) = 1 | + (−0.886 + 1.53i)2-s + (−1.07 − 1.85i)4-s + (−0.223 + 0.387i)5-s − 0.0838·7-s + 2.02·8-s + (−0.396 − 0.686i)10-s + 0.234·11-s + (0.693 + 1.20i)13-s + (0.0743 − 0.128i)14-s + (−0.724 + 1.25i)16-s + (0.858 − 1.48i)17-s + (0.305 − 0.952i)19-s + 0.958·20-s + (−0.207 + 0.360i)22-s + (0.842 + 1.45i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 855 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.813 - 0.582i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 855 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.813 - 0.582i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.263141 + 0.819373i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.263141 + 0.819373i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 5 | \( 1 + (0.5 - 0.866i)T \) |
| 19 | \( 1 + (-1.33 + 4.15i)T \) |
good | 2 | \( 1 + (1.25 - 2.17i)T + (-1 - 1.73i)T^{2} \) |
| 7 | \( 1 + 0.221T + 7T^{2} \) |
| 11 | \( 1 - 0.778T + 11T^{2} \) |
| 13 | \( 1 + (-2.5 - 4.33i)T + (-6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 + (-3.53 + 6.12i)T + (-8.5 - 14.7i)T^{2} \) |
| 23 | \( 1 + (-4.03 - 6.99i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (-0.110 - 0.192i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 - 2.50T + 31T^{2} \) |
| 37 | \( 1 + 1.90T + 37T^{2} \) |
| 41 | \( 1 + (3.61 - 6.26i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (3.64 - 6.32i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (1.39 + 2.41i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (-2.19 - 3.79i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (1.39 - 2.41i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-6.29 - 10.8i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-5.28 - 9.15i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (4.92 - 8.52i)T + (-35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + (-7.03 + 12.1i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (0.792 - 1.37i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 - 9.52T + 83T^{2} \) |
| 89 | \( 1 + (-1.57 - 2.71i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (-3.18 + 5.51i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−10.02353205639661301890981016913, −9.409236850473198794405353817714, −8.815537272026923827648681741600, −7.80621779225656142920429052226, −7.06163146215140943796205668627, −6.60117903658806493246435335726, −5.50854718627252836350811122878, −4.66272543541607611111443837977, −3.17455059096047881226755744440, −1.14564056617295789954246641625,
0.72094563299972330615224213059, 1.83195136494491115977167745430, 3.28139273191288228166320499463, 3.80721491750881589511302228519, 5.19207595269564458102058354266, 6.41494137843336007702231767322, 8.006144619620775601142410160153, 8.251802364272967602692057660928, 9.099204150865725331147498206569, 10.19840043327876221808514081397