L(s) = 1 | + (1.31 + 1.12i)3-s + 5-s + (−2.64 − 0.0644i)7-s + (0.468 + 2.96i)9-s − 2.69i·11-s + 6.61i·13-s + (1.31 + 1.12i)15-s − 7.03·17-s + 3.22i·19-s + (−3.41 − 3.06i)21-s − 2.56i·23-s + 25-s + (−2.71 + 4.42i)27-s + 8.21i·29-s − 2.87i·31-s + ⋯ |
L(s) = 1 | + (0.760 + 0.649i)3-s + 0.447·5-s + (−0.999 − 0.0243i)7-s + (0.156 + 0.987i)9-s − 0.813i·11-s + 1.83i·13-s + (0.340 + 0.290i)15-s − 1.70·17-s + 0.740i·19-s + (−0.744 − 0.667i)21-s − 0.534i·23-s + 0.200·25-s + (−0.522 + 0.852i)27-s + 1.52i·29-s − 0.515i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1680 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.744 - 0.667i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1680 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.744 - 0.667i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.433320858\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.433320858\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (-1.31 - 1.12i)T \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 + (2.64 + 0.0644i)T \) |
good | 11 | \( 1 + 2.69iT - 11T^{2} \) |
| 13 | \( 1 - 6.61iT - 13T^{2} \) |
| 17 | \( 1 + 7.03T + 17T^{2} \) |
| 19 | \( 1 - 3.22iT - 19T^{2} \) |
| 23 | \( 1 + 2.56iT - 23T^{2} \) |
| 29 | \( 1 - 8.21iT - 29T^{2} \) |
| 31 | \( 1 + 2.87iT - 31T^{2} \) |
| 37 | \( 1 + 6.79T + 37T^{2} \) |
| 41 | \( 1 - 10.0T + 41T^{2} \) |
| 43 | \( 1 + 11.6T + 43T^{2} \) |
| 47 | \( 1 - 2.56T + 47T^{2} \) |
| 53 | \( 1 - 2.70iT - 53T^{2} \) |
| 59 | \( 1 - 3.03T + 59T^{2} \) |
| 61 | \( 1 - 7.83iT - 61T^{2} \) |
| 67 | \( 1 - 12.5T + 67T^{2} \) |
| 71 | \( 1 - 14.0iT - 71T^{2} \) |
| 73 | \( 1 + 1.81iT - 73T^{2} \) |
| 79 | \( 1 + 13.4T + 79T^{2} \) |
| 83 | \( 1 + 3.01T + 83T^{2} \) |
| 89 | \( 1 - 4.17T + 89T^{2} \) |
| 97 | \( 1 + 9.55iT - 97T^{2} \) |
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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
−9.492076800645596774633124638689, −8.888591140559419575654133212723, −8.524085306254015181226405753559, −7.07123314395084927464734180734, −6.59574895009219809895165802183, −5.59501264465538863346861485730, −4.44854279614245171569461413082, −3.80074885494852962548158468163, −2.77144697503171336405987574327, −1.83917378730064954363975644986,
0.45116903034907102683641806009, 2.07512388545733492385405712405, 2.81268971547573920024670663925, 3.73248233813783366374128344149, 4.94866701125245173754183193394, 6.04298400789088867570166450301, 6.71744664935922107233587895770, 7.40508721437889358538173451903, 8.290283786678857977793232844583, 9.052816274498722634835994959167