L(s) = 1 | + (−0.5 − 0.866i)2-s + (0.619 + 1.61i)3-s + (−0.499 + 0.866i)4-s + 1.76·5-s + (1.09 − 1.34i)6-s + (−1.85 + 1.88i)7-s + 0.999·8-s + (−2.23 + 2.00i)9-s + (−0.880 − 1.52i)10-s + 6.12·11-s + (−1.71 − 0.272i)12-s + (−0.380 − 0.658i)13-s + (2.56 + 0.658i)14-s + (1.09 + 2.84i)15-s + (−0.5 − 0.866i)16-s + (−3.42 − 5.92i)17-s + ⋯ |
L(s) = 1 | + (−0.353 − 0.612i)2-s + (0.357 + 0.933i)3-s + (−0.249 + 0.433i)4-s + 0.787·5-s + (0.445 − 0.549i)6-s + (−0.699 + 0.714i)7-s + 0.353·8-s + (−0.744 + 0.668i)9-s + (−0.278 − 0.482i)10-s + 1.84·11-s + (−0.493 − 0.0785i)12-s + (−0.105 − 0.182i)13-s + (0.684 + 0.176i)14-s + (0.281 + 0.735i)15-s + (−0.125 − 0.216i)16-s + (−0.829 − 1.43i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 126 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.932 - 0.360i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 126 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.932 - 0.360i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.03286 + 0.192623i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.03286 + 0.192623i\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + (-0.619 - 1.61i)T \) |
| 7 | \( 1 + (1.85 - 1.88i)T \) |
good | 5 | \( 1 - 1.76T + 5T^{2} \) |
| 11 | \( 1 - 6.12T + 11T^{2} \) |
| 13 | \( 1 + (0.380 + 0.658i)T + (-6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 + (3.42 + 5.92i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-0.971 + 1.68i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + 0.421T + 23T^{2} \) |
| 29 | \( 1 + (-0.732 + 1.26i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (3.85 - 6.67i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (-1.44 + 2.49i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (3.47 + 6.01i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (-4.33 + 7.49i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (0.830 + 1.43i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (0.112 + 0.195i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (0.993 - 1.72i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-5.17 - 8.96i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (3.39 - 5.87i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 - 10.7T + 71T^{2} \) |
| 73 | \( 1 + (-0.153 - 0.265i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (-6.72 - 11.6i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (1.56 - 2.70i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + (-1.30 + 2.25i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (1.81 - 3.14i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−13.58241344239110627243981851791, −12.17876662681443053718114506039, −11.32250453452765251182154822687, −10.05829639017993483027583967290, −9.205997770174821808102695185454, −8.914528273159782645156597601528, −6.84586009112041544798600984082, −5.38156402585809942558652171557, −3.82602350489586755864695107132, −2.43264635136228650574994170717,
1.60239157221177772127524824236, 3.88873377960504604175660913491, 6.24704939071819496207087011450, 6.49010363556043403578133744722, 7.84300079451332610282720966480, 9.094256753027366420301790684643, 9.772892219903700947753601135788, 11.26849199028467323822118719503, 12.62213775704383326477558567447, 13.47617590443718861434047158303