L(s) = 1 | + (0.750 − 1.19i)2-s + (0.448 − 1.67i)3-s + (−0.874 − 1.79i)4-s + 3.56i·5-s + (−1.66 − 1.79i)6-s + i·7-s + (−2.81 − 0.301i)8-s + (−2.59 − 1.50i)9-s + (4.27 + 2.67i)10-s − 0.335·11-s + (−3.40 + 0.655i)12-s + 3.34·13-s + (1.19 + 0.750i)14-s + (5.96 + 1.59i)15-s + (−2.47 + 3.14i)16-s + 0.335i·17-s + ⋯ |
L(s) = 1 | + (0.530 − 0.847i)2-s + (0.258 − 0.965i)3-s + (−0.437 − 0.899i)4-s + 1.59i·5-s + (−0.681 − 0.731i)6-s + 0.377i·7-s + (−0.994 − 0.106i)8-s + (−0.865 − 0.500i)9-s + (1.35 + 0.845i)10-s − 0.101·11-s + (−0.981 + 0.189i)12-s + 0.928·13-s + (0.320 + 0.200i)14-s + (1.53 + 0.412i)15-s + (−0.617 + 0.786i)16-s + 0.0813i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 84 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.189 + 0.981i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 84 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.189 + 0.981i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.940017 - 0.776134i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.940017 - 0.776134i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.750 + 1.19i)T \) |
| 3 | \( 1 + (-0.448 + 1.67i)T \) |
| 7 | \( 1 - iT \) |
good | 5 | \( 1 - 3.56iT - 5T^{2} \) |
| 11 | \( 1 + 0.335T + 11T^{2} \) |
| 13 | \( 1 - 3.34T + 13T^{2} \) |
| 17 | \( 1 - 0.335iT - 17T^{2} \) |
| 19 | \( 1 + 1.84iT - 19T^{2} \) |
| 23 | \( 1 + 4.45T + 23T^{2} \) |
| 29 | \( 1 + 5.91iT - 29T^{2} \) |
| 31 | \( 1 - 5.19iT - 31T^{2} \) |
| 37 | \( 1 - 3.19T + 37T^{2} \) |
| 41 | \( 1 - 1.45iT - 41T^{2} \) |
| 43 | \( 1 + 7.49iT - 43T^{2} \) |
| 47 | \( 1 + 8.91T + 47T^{2} \) |
| 53 | \( 1 + 4.79iT - 53T^{2} \) |
| 59 | \( 1 - 14.0T + 59T^{2} \) |
| 61 | \( 1 + 0.353T + 61T^{2} \) |
| 67 | \( 1 - 3.19iT - 67T^{2} \) |
| 71 | \( 1 - 10.3T + 71T^{2} \) |
| 73 | \( 1 + 4.69T + 73T^{2} \) |
| 79 | \( 1 - 4iT - 79T^{2} \) |
| 83 | \( 1 + 6.89T + 83T^{2} \) |
| 89 | \( 1 + 3.87iT - 89T^{2} \) |
| 97 | \( 1 + 2T + 97T^{2} \) |
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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
−13.93355871930954896110904918693, −13.06993669972082585047983374178, −11.79501487953859952550238008702, −11.11037154355871987431109699800, −9.939109350314260444833224335925, −8.398963881976072422425420213824, −6.81280509332250119822869063759, −5.89945417986949285636535797622, −3.49805639428631001786748240718, −2.28496944310686873363744545772,
3.80992266774595779152159052255, 4.80676815139854648064208049283, 5.91299378196960688818271448792, 7.975717431978972952439262347030, 8.722520243832338984390305706054, 9.754826164854288600859256925489, 11.44737623021923705938010315861, 12.73017392197344045234600496747, 13.56531951055440061903874008996, 14.56746585428248174187061042756