L(s) = 1 | − 3.46i·5-s − 2.64·7-s + 4.58i·13-s − 1.73i·17-s − 5.29·19-s + 4.58i·23-s − 6.99·25-s + 7.93·29-s − 2.64·31-s + 9.16i·35-s + 4·37-s + 6.92i·41-s − 1.73i·43-s − 6·47-s + 7.00·49-s + ⋯ |
L(s) = 1 | − 1.54i·5-s − 0.999·7-s + 1.27i·13-s − 0.420i·17-s − 1.21·19-s + 0.955i·23-s − 1.39·25-s + 1.47·29-s − 0.475·31-s + 1.54i·35-s + 0.657·37-s + 1.08i·41-s − 0.264i·43-s − 0.875·47-s + 49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.5 - 0.866i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.5 - 0.866i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.8776394971\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8776394971\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + 2.64T \) |
good | 5 | \( 1 + 3.46iT - 5T^{2} \) |
| 11 | \( 1 - 11T^{2} \) |
| 13 | \( 1 - 4.58iT - 13T^{2} \) |
| 17 | \( 1 + 1.73iT - 17T^{2} \) |
| 19 | \( 1 + 5.29T + 19T^{2} \) |
| 23 | \( 1 - 4.58iT - 23T^{2} \) |
| 29 | \( 1 - 7.93T + 29T^{2} \) |
| 31 | \( 1 + 2.64T + 31T^{2} \) |
| 37 | \( 1 - 4T + 37T^{2} \) |
| 41 | \( 1 - 6.92iT - 41T^{2} \) |
| 43 | \( 1 + 1.73iT - 43T^{2} \) |
| 47 | \( 1 + 6T + 47T^{2} \) |
| 53 | \( 1 + 7.93T + 53T^{2} \) |
| 59 | \( 1 + 3T + 59T^{2} \) |
| 61 | \( 1 - 9.16iT - 61T^{2} \) |
| 67 | \( 1 - 12.1iT - 67T^{2} \) |
| 71 | \( 1 - 4.58iT - 71T^{2} \) |
| 73 | \( 1 + 9.16iT - 73T^{2} \) |
| 79 | \( 1 + 6.92iT - 79T^{2} \) |
| 83 | \( 1 - 12T + 83T^{2} \) |
| 89 | \( 1 - 5.19iT - 89T^{2} \) |
| 97 | \( 1 + 9.16iT - 97T^{2} \) |
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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
−8.879374791507878557709670929902, −8.335623516144387085833434931604, −7.33935125088105743776056311064, −6.47870380945965778313310664847, −5.88057113563610627400037320937, −4.76335103094274571025763720278, −4.38317100606364102124423469455, −3.33700483409131087015183657450, −2.10826812791739662003575078402, −1.02634958463653788423674866876,
0.30996924983463594574558091552, 2.21486702787989740790102378284, 2.99692983859679776130038916266, 3.54581441697250822623071516613, 4.63215516681622433331001492695, 5.87370656747971866759144125921, 6.42716797466483306966862560216, 6.86683169056934973055324779701, 7.84670087808327420889141006231, 8.455098356724067382512696698516