L(s) = 1 | + (0.5 − 0.866i)2-s + (−1.5 + 0.866i)3-s + (−0.499 − 0.866i)4-s + (1.5 + 2.59i)5-s + 1.73i·6-s − 0.999·8-s + (1.5 − 2.59i)9-s + 3·10-s + (1.5 − 2.59i)11-s + (1.49 + 0.866i)12-s + (−0.5 − 0.866i)13-s + (−4.5 − 2.59i)15-s + (−0.5 + 0.866i)16-s − 3·17-s + (−1.5 − 2.59i)18-s + 7·19-s + ⋯ |
L(s) = 1 | + (0.353 − 0.612i)2-s + (−0.866 + 0.499i)3-s + (−0.249 − 0.433i)4-s + (0.670 + 1.16i)5-s + 0.707i·6-s − 0.353·8-s + (0.5 − 0.866i)9-s + 0.948·10-s + (0.452 − 0.783i)11-s + (0.433 + 0.250i)12-s + (−0.138 − 0.240i)13-s + (−1.16 − 0.670i)15-s + (−0.125 + 0.216i)16-s − 0.727·17-s + (−0.353 − 0.612i)18-s + 1.60·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.939 - 0.342i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.939 - 0.342i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.55024 + 0.273350i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.55024 + 0.273350i\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + (1.5 - 0.866i)T \) |
| 7 | \( 1 \) |
good | 5 | \( 1 + (-1.5 - 2.59i)T + (-2.5 + 4.33i)T^{2} \) |
| 11 | \( 1 + (-1.5 + 2.59i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (0.5 + 0.866i)T + (-6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 + 3T + 17T^{2} \) |
| 19 | \( 1 - 7T + 19T^{2} \) |
| 23 | \( 1 + (-4.5 - 7.79i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (1.5 - 2.59i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (-4 - 6.92i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + T + 37T^{2} \) |
| 41 | \( 1 + (-1.5 - 2.59i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (-0.5 + 0.866i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 - 3T + 53T^{2} \) |
| 59 | \( 1 + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-1 + 1.73i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-2 - 3.46i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 - 12T + 71T^{2} \) |
| 73 | \( 1 + 11T + 73T^{2} \) |
| 79 | \( 1 + (-8 + 13.8i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (4.5 - 7.79i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + 3T + 89T^{2} \) |
| 97 | \( 1 + (0.5 - 0.866i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−10.31890396833786160270954866036, −9.673463905284701070644229080629, −8.936261980336884443152273871707, −7.28306231394502376893488186712, −6.54754633643403299961289665079, −5.67476906995538082895750696529, −5.01123085682411587890281074804, −3.59513487770469630493419161079, −2.96324445485016578225124427418, −1.27136214522902862082356429554,
0.908358593200701456434204711347, 2.28667728761921153767464113895, 4.30918883849380669987224604026, 4.91636534811541873833526297319, 5.68671321128062997522276327147, 6.57704481571504673144800439881, 7.29117996637768372246160041753, 8.301183231056055855344608894092, 9.255505984517883432505675079881, 9.877447480455617376993334873529