L(s) = 1 | + (1.15 + 0.818i)2-s + (0.660 + 1.88i)4-s − 2.16i·5-s + (1.94 + 1.79i)7-s + (−0.782 + 2.71i)8-s + (1.76 − 2.49i)10-s + 0.414i·11-s + 2.36i·13-s + (0.782 + 3.65i)14-s + (−3.12 + 2.49i)16-s + 0.695i·17-s + 5.21·19-s + (4.08 − 1.42i)20-s + (−0.339 + 0.478i)22-s − 0.414i·23-s + ⋯ |
L(s) = 1 | + (0.815 + 0.578i)2-s + (0.330 + 0.943i)4-s − 0.967i·5-s + (0.736 + 0.676i)7-s + (−0.276 + 0.960i)8-s + (0.559 − 0.788i)10-s + 0.124i·11-s + 0.656i·13-s + (0.209 + 0.977i)14-s + (−0.781 + 0.623i)16-s + 0.168i·17-s + 1.19·19-s + (0.912 − 0.319i)20-s + (−0.0723 + 0.101i)22-s − 0.0864i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.395 - 0.918i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.395 - 0.918i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.18249 + 1.43683i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.18249 + 1.43683i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-1.15 - 0.818i)T \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (-1.94 - 1.79i)T \) |
good | 5 | \( 1 + 2.16iT - 5T^{2} \) |
| 11 | \( 1 - 0.414iT - 11T^{2} \) |
| 13 | \( 1 - 2.36iT - 13T^{2} \) |
| 17 | \( 1 - 0.695iT - 17T^{2} \) |
| 19 | \( 1 - 5.21T + 19T^{2} \) |
| 23 | \( 1 + 0.414iT - 23T^{2} \) |
| 29 | \( 1 - 9.73T + 29T^{2} \) |
| 31 | \( 1 + 2.03T + 31T^{2} \) |
| 37 | \( 1 + 7.43T + 37T^{2} \) |
| 41 | \( 1 + 10.5iT - 41T^{2} \) |
| 43 | \( 1 - 8.76iT - 43T^{2} \) |
| 47 | \( 1 + 5.11T + 47T^{2} \) |
| 53 | \( 1 - 3.12T + 53T^{2} \) |
| 59 | \( 1 + 11.2T + 59T^{2} \) |
| 61 | \( 1 + 3.97iT - 61T^{2} \) |
| 67 | \( 1 - 2.36iT - 67T^{2} \) |
| 71 | \( 1 + 4.04iT - 71T^{2} \) |
| 73 | \( 1 + 12.3iT - 73T^{2} \) |
| 79 | \( 1 + 7.61iT - 79T^{2} \) |
| 83 | \( 1 + 12.8T + 83T^{2} \) |
| 89 | \( 1 + 14.6iT - 89T^{2} \) |
| 97 | \( 1 - 3.97iT - 97T^{2} \) |
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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
−10.68593898800873536906772087346, −9.287524575976850977070169669847, −8.658588499015753657283865301101, −7.927826859001230875278532015879, −6.94626849229568317716031085858, −5.87823386555576700813810877775, −5.00785368479160170449716575430, −4.50145746333776382335419151048, −3.11331484324358116240431673539, −1.70177444763141456543551710998,
1.20462648073589392570416876371, 2.73316976755688573725253221814, 3.50712056423131877847108422611, 4.68860640349910671789472617297, 5.51741112154044923965466695430, 6.67848058726480090425829884858, 7.31684647331367963057766345114, 8.413924530670719636917125526922, 9.810239650276179980024040885505, 10.39348252544085791004781907343