L(s) = 1 | + (1.35 + 1.13i)2-s + (−1.86 + 0.677i)3-s + (0.367 + 2.08i)4-s + (−3.28 − 1.19i)6-s + (−0.719 − 1.24i)7-s + (−0.987 + 1.71i)8-s + (2.23 − 1.87i)9-s + (0.5 − 0.866i)11-s + (−2.09 − 3.63i)12-s + (0.939 + 0.342i)13-s + (0.441 − 2.50i)14-s + (−1.28 + 0.468i)16-s + 5.16·18-s + (0.615 − 0.788i)19-s + (2.18 + 1.83i)21-s + (1.65 − 0.603i)22-s + ⋯ |
L(s) = 1 | + (1.35 + 1.13i)2-s + (−1.86 + 0.677i)3-s + (0.367 + 2.08i)4-s + (−3.28 − 1.19i)6-s + (−0.719 − 1.24i)7-s + (−0.987 + 1.71i)8-s + (2.23 − 1.87i)9-s + (0.5 − 0.866i)11-s + (−2.09 − 3.63i)12-s + (0.939 + 0.342i)13-s + (0.441 − 2.50i)14-s + (−1.28 + 0.468i)16-s + 5.16·18-s + (0.615 − 0.788i)19-s + (2.18 + 1.83i)21-s + (1.65 − 0.603i)22-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2717 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.603 - 0.797i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2717 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.603 - 0.797i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.295876357\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.295876357\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 11 | \( 1 + (-0.5 + 0.866i)T \) |
| 13 | \( 1 + (-0.939 - 0.342i)T \) |
| 19 | \( 1 + (-0.615 + 0.788i)T \) |
good | 2 | \( 1 + (-1.35 - 1.13i)T + (0.173 + 0.984i)T^{2} \) |
| 3 | \( 1 + (1.86 - 0.677i)T + (0.766 - 0.642i)T^{2} \) |
| 5 | \( 1 + (0.939 + 0.342i)T^{2} \) |
| 7 | \( 1 + (0.719 + 1.24i)T + (-0.5 + 0.866i)T^{2} \) |
| 17 | \( 1 + (-0.173 - 0.984i)T^{2} \) |
| 23 | \( 1 + (0.346 + 1.96i)T + (-0.939 + 0.342i)T^{2} \) |
| 29 | \( 1 + (-0.173 + 0.984i)T^{2} \) |
| 31 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 37 | \( 1 - T^{2} \) |
| 41 | \( 1 + (0.454 - 0.165i)T + (0.766 - 0.642i)T^{2} \) |
| 43 | \( 1 + (0.939 + 0.342i)T^{2} \) |
| 47 | \( 1 + (-0.173 + 0.984i)T^{2} \) |
| 53 | \( 1 + (0.130 + 0.737i)T + (-0.939 + 0.342i)T^{2} \) |
| 59 | \( 1 + (-0.173 - 0.984i)T^{2} \) |
| 61 | \( 1 + (0.939 - 0.342i)T^{2} \) |
| 67 | \( 1 + (-0.173 + 0.984i)T^{2} \) |
| 71 | \( 1 + (0.939 + 0.342i)T^{2} \) |
| 73 | \( 1 + (-0.580 + 0.211i)T + (0.766 - 0.642i)T^{2} \) |
| 79 | \( 1 + (-0.766 + 0.642i)T^{2} \) |
| 83 | \( 1 + (-0.438 - 0.759i)T + (-0.5 + 0.866i)T^{2} \) |
| 89 | \( 1 + (-0.766 - 0.642i)T^{2} \) |
| 97 | \( 1 + (-0.173 - 0.984i)T^{2} \) |
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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
−9.147778531546102548164613038512, −7.993742600874135373443955999843, −6.86250015270898700346805591308, −6.60397266288087223140161576602, −6.11538097015229498595614891105, −5.34103706015611914228592190898, −4.48114440322735074444760190725, −3.99823293667939082587734706051, −3.39160415883714317957118840310, −0.74762704529894936654153735946,
1.40462800934705747054331333373, 1.96724413648700352812289084797, 3.35944607535613157255147338568, 4.19648953414862425483425817962, 5.22696502149361391090182721559, 5.75329655200092180740581848534, 6.01453324195178497880799551865, 6.92016303887971069709140455955, 7.86148483559458631606592568675, 9.484820676128452592964098639349