L(s) = 1 | + (0.5 + 0.866i)2-s + (1.5 − 0.866i)3-s + (−0.499 + 0.866i)4-s + (−1 + 1.73i)5-s + (1.5 + 0.866i)6-s + (0.5 + 0.866i)7-s − 0.999·8-s + (1.5 − 2.59i)9-s − 1.99·10-s + (−0.5 − 0.866i)11-s + 1.73i·12-s + (3 − 5.19i)13-s + (−0.499 + 0.866i)14-s + 3.46i·15-s + (−0.5 − 0.866i)16-s − 5·17-s + ⋯ |
L(s) = 1 | + (0.353 + 0.612i)2-s + (0.866 − 0.499i)3-s + (−0.249 + 0.433i)4-s + (−0.447 + 0.774i)5-s + (0.612 + 0.353i)6-s + (0.188 + 0.327i)7-s − 0.353·8-s + (0.5 − 0.866i)9-s − 0.632·10-s + (−0.150 − 0.261i)11-s + 0.499i·12-s + (0.832 − 1.44i)13-s + (−0.133 + 0.231i)14-s + 0.894i·15-s + (−0.125 − 0.216i)16-s − 1.21·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 126 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.766 - 0.642i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 126 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.766 - 0.642i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.37693 + 0.501164i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.37693 + 0.501164i\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + (-0.5 - 0.866i)T \) |
good | 5 | \( 1 + (1 - 1.73i)T + (-2.5 - 4.33i)T^{2} \) |
| 11 | \( 1 + (0.5 + 0.866i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (-3 + 5.19i)T + (-6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 + 5T + 17T^{2} \) |
| 19 | \( 1 + 7T + 19T^{2} \) |
| 23 | \( 1 + (2 - 3.46i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (-2 - 3.46i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (-3 + 5.19i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 - 2T + 37T^{2} \) |
| 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 | \( 1 - 12T + 53T^{2} \) |
| 59 | \( 1 + (-3.5 + 6.06i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-6 - 10.3i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (6.5 - 11.2i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + 8T + 71T^{2} \) |
| 73 | \( 1 - T + 73T^{2} \) |
| 79 | \( 1 + (-3 - 5.19i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (8 + 13.8i)T + (-41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + 6T + 89T^{2} \) |
| 97 | \( 1 + (-2.5 - 4.33i)T + (-48.5 + 84.0i)T^{2} \) |
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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
−13.39560702532095048814899539897, −12.97155915050280027369317266507, −11.56602850860440395960291097433, −10.42400442228366273439856855010, −8.756128605346644803185162649711, −8.112813899750194172499269198264, −7.00045178256891357249556825515, −5.95044321299502134099436925272, −4.01045342884639869263048573210, −2.73566829821513854369043767683,
2.12785420891147681655508486085, 4.14404042175073924164565894554, 4.52480945873364556384129355616, 6.64908300524279367363703606192, 8.441339527570770070614362827090, 8.876711856484510325326696871937, 10.24557188885592374799801775324, 11.15660330935040008044317610864, 12.34812941571619881299164716439, 13.34847133560803116986308430748