| L(s) = 1 | − 0.559i·2-s + 1.68·4-s + 2.18i·5-s − 3.09i·7-s − 2.06i·8-s + 1.22·10-s − 0.559i·11-s + (2.90 + 2.12i)13-s − 1.73·14-s + 2.22·16-s + 3.30·17-s − 0.688i·19-s + 3.68i·20-s − 0.312·22-s − 2.11·23-s + ⋯ |
| L(s) = 1 | − 0.395i·2-s + 0.843·4-s + 0.977i·5-s − 1.17i·7-s − 0.728i·8-s + 0.386·10-s − 0.168i·11-s + (0.806 + 0.590i)13-s − 0.462·14-s + 0.555·16-s + 0.802·17-s − 0.157i·19-s + 0.824i·20-s − 0.0666·22-s − 0.441·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1053 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.806 + 0.590i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1053 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.806 + 0.590i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.126605428\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.126605428\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 13 | \( 1 + (-2.90 - 2.12i)T \) |
| good | 2 | \( 1 + 0.559iT - 2T^{2} \) |
| 5 | \( 1 - 2.18iT - 5T^{2} \) |
| 7 | \( 1 + 3.09iT - 7T^{2} \) |
| 11 | \( 1 + 0.559iT - 11T^{2} \) |
| 17 | \( 1 - 3.30T + 17T^{2} \) |
| 19 | \( 1 + 0.688iT - 19T^{2} \) |
| 23 | \( 1 + 2.11T + 23T^{2} \) |
| 29 | \( 1 + 3.84T + 29T^{2} \) |
| 31 | \( 1 + 6.66iT - 31T^{2} \) |
| 37 | \( 1 + 5.91iT - 37T^{2} \) |
| 41 | \( 1 - 7.42iT - 41T^{2} \) |
| 43 | \( 1 - 3.46T + 43T^{2} \) |
| 47 | \( 1 - 10.2iT - 47T^{2} \) |
| 53 | \( 1 - 14.3T + 53T^{2} \) |
| 59 | \( 1 - 4.19iT - 59T^{2} \) |
| 61 | \( 1 + 2.31T + 61T^{2} \) |
| 67 | \( 1 + 5.72iT - 67T^{2} \) |
| 71 | \( 1 + 16.0iT - 71T^{2} \) |
| 73 | \( 1 - 15.8iT - 73T^{2} \) |
| 79 | \( 1 - 1.28T + 79T^{2} \) |
| 83 | \( 1 + 8.42iT - 83T^{2} \) |
| 89 | \( 1 + 6.99iT - 89T^{2} \) |
| 97 | \( 1 - 9.89iT - 97T^{2} \) |
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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
−10.13183476725521803547127506184, −9.247786834810793161119167830635, −7.81470497987804110194354740596, −7.34733261499230640343958282091, −6.51322332780445174827899826198, −5.84578974163524987665987301654, −4.14716151799529893181323130292, −3.49481685988126692689569622977, −2.44839645983529815226697564299, −1.13435173846404642271632273995,
1.36164990581027936881305952796, 2.52987403247356582018289234884, 3.70553970237139751092220977804, 5.33806724394332599236382654272, 5.49673657274705610182092123581, 6.54266457413703949399590583362, 7.55332157516560662020317508527, 8.494009209605531837023812425399, 8.786648039911350841831331774413, 9.993574711119466225687645324197
