L(s) = 1 | + (0.272 − 1.38i)2-s + (−1.85 − 0.755i)4-s + (−2.12 − 1.22i)5-s + (−2.63 + 0.272i)7-s + (−1.55 + 2.36i)8-s + (−2.27 + 2.61i)10-s + (−1.09 + 0.632i)11-s − 2.99i·13-s + (−0.337 + 3.72i)14-s + (2.85 + 2.79i)16-s + (−1.58 + 0.916i)17-s + (2.07 − 3.60i)19-s + (3.00 + 3.87i)20-s + (0.579 + 1.69i)22-s + (−5.83 − 3.36i)23-s + ⋯ |
L(s) = 1 | + (0.192 − 0.981i)2-s + (−0.925 − 0.377i)4-s + (−0.949 − 0.548i)5-s + (−0.994 + 0.102i)7-s + (−0.548 + 0.836i)8-s + (−0.720 + 0.826i)10-s + (−0.330 + 0.190i)11-s − 0.831i·13-s + (−0.0903 + 0.995i)14-s + (0.714 + 0.699i)16-s + (−0.385 + 0.222i)17-s + (0.477 − 0.826i)19-s + (0.672 + 0.866i)20-s + (0.123 + 0.360i)22-s + (−1.21 − 0.702i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.916 - 0.399i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.916 - 0.399i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.101066 + 0.485164i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.101066 + 0.485164i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.272 + 1.38i)T \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (2.63 - 0.272i)T \) |
good | 5 | \( 1 + (2.12 + 1.22i)T + (2.5 + 4.33i)T^{2} \) |
| 11 | \( 1 + (1.09 - 0.632i)T + (5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + 2.99iT - 13T^{2} \) |
| 17 | \( 1 + (1.58 - 0.916i)T + (8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-2.07 + 3.60i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (5.83 + 3.36i)T + (11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 - 9.42T + 29T^{2} \) |
| 31 | \( 1 + (4.71 + 8.17i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (3.75 - 6.50i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + 1.08iT - 41T^{2} \) |
| 43 | \( 1 - 6.27iT - 43T^{2} \) |
| 47 | \( 1 + (-3.67 + 6.36i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (0.0358 + 0.0620i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (-1.68 - 2.91i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (9.61 + 5.55i)T + (30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-2.43 + 1.40i)T + (33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + 2.92iT - 71T^{2} \) |
| 73 | \( 1 + (-7.01 + 4.05i)T + (36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-1.54 - 0.891i)T + (39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + 5.33T + 83T^{2} \) |
| 89 | \( 1 + (7.42 + 4.28i)T + (44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 - 7.10iT - 97T^{2} \) |
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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
−11.72076070559852928424724091714, −10.56587853171830758620526164862, −9.794592486501520656437337906476, −8.712535651478265908503582152224, −7.85724056616243720316139160257, −6.25783644937171220415123436576, −4.91768762226585871039423192228, −3.87650682769617184217023651112, −2.68428477137963704997372154108, −0.36219522303963726007297558965,
3.27633804344056507388961554465, 4.15709167695639846963672137177, 5.66402505474332467439114834058, 6.76973829465164327394961577188, 7.43014105906402848764767731902, 8.502431564270471196645562160477, 9.539985890528176805946359143660, 10.56954974110546945412998324922, 11.92147985163243007518655746195, 12.54291407936011278431211659870