| L(s) = 1 | + (−1.12 − 0.854i)2-s + (−2.15 − 0.212i)3-s + (0.540 + 1.92i)4-s + (−0.471 − 0.881i)5-s + (2.24 + 2.07i)6-s + (−0.288 − 1.44i)7-s + (1.03 − 2.63i)8-s + (1.65 + 0.328i)9-s + (−0.221 + 1.39i)10-s + (3.98 + 3.27i)11-s + (−0.756 − 4.26i)12-s + (−1.20 − 0.646i)13-s + (−0.912 + 1.87i)14-s + (0.828 + 1.99i)15-s + (−3.41 + 2.08i)16-s + (−1.15 + 2.79i)17-s + ⋯ |
| L(s) = 1 | + (−0.797 − 0.603i)2-s + (−1.24 − 0.122i)3-s + (0.270 + 0.962i)4-s + (−0.210 − 0.394i)5-s + (0.916 + 0.848i)6-s + (−0.108 − 0.547i)7-s + (0.365 − 0.930i)8-s + (0.550 + 0.109i)9-s + (−0.0701 + 0.441i)10-s + (1.20 + 0.986i)11-s + (−0.218 − 1.23i)12-s + (−0.335 − 0.179i)13-s + (−0.243 + 0.502i)14-s + (0.213 + 0.516i)15-s + (−0.853 + 0.520i)16-s + (−0.281 + 0.678i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 640 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.688 + 0.725i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 640 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.688 + 0.725i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.181830 - 0.423352i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.181830 - 0.423352i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.12 + 0.854i)T \) |
| 5 | \( 1 + (0.471 + 0.881i)T \) |
| good | 3 | \( 1 + (2.15 + 0.212i)T + (2.94 + 0.585i)T^{2} \) |
| 7 | \( 1 + (0.288 + 1.44i)T + (-6.46 + 2.67i)T^{2} \) |
| 11 | \( 1 + (-3.98 - 3.27i)T + (2.14 + 10.7i)T^{2} \) |
| 13 | \( 1 + (1.20 + 0.646i)T + (7.22 + 10.8i)T^{2} \) |
| 17 | \( 1 + (1.15 - 2.79i)T + (-12.0 - 12.0i)T^{2} \) |
| 19 | \( 1 + (-7.24 + 2.19i)T + (15.7 - 10.5i)T^{2} \) |
| 23 | \( 1 + (-1.64 + 1.09i)T + (8.80 - 21.2i)T^{2} \) |
| 29 | \( 1 + (5.91 + 7.20i)T + (-5.65 + 28.4i)T^{2} \) |
| 31 | \( 1 + (4.70 - 4.70i)T - 31iT^{2} \) |
| 37 | \( 1 + (-3.28 + 10.8i)T + (-30.7 - 20.5i)T^{2} \) |
| 41 | \( 1 + (6.08 + 9.11i)T + (-15.6 + 37.8i)T^{2} \) |
| 43 | \( 1 + (-0.174 + 0.0171i)T + (42.1 - 8.38i)T^{2} \) |
| 47 | \( 1 + (8.95 + 3.71i)T + (33.2 + 33.2i)T^{2} \) |
| 53 | \( 1 + (5.24 - 6.39i)T + (-10.3 - 51.9i)T^{2} \) |
| 59 | \( 1 + (-7.40 + 3.95i)T + (32.7 - 49.0i)T^{2} \) |
| 61 | \( 1 + (0.103 - 1.04i)T + (-59.8 - 11.9i)T^{2} \) |
| 67 | \( 1 + (-0.270 + 2.74i)T + (-65.7 - 13.0i)T^{2} \) |
| 71 | \( 1 + (5.60 - 1.11i)T + (65.5 - 27.1i)T^{2} \) |
| 73 | \( 1 + (1.50 - 7.55i)T + (-67.4 - 27.9i)T^{2} \) |
| 79 | \( 1 + (-2.81 + 1.16i)T + (55.8 - 55.8i)T^{2} \) |
| 83 | \( 1 + (4.75 + 15.6i)T + (-69.0 + 46.1i)T^{2} \) |
| 89 | \( 1 + (10.8 + 7.24i)T + (34.0 + 82.2i)T^{2} \) |
| 97 | \( 1 + (-10.1 + 10.1i)T - 97iT^{2} \) |
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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
−10.30030498632346966317955561308, −9.541793110232235276921984189797, −8.764451368090759570145493784993, −7.33850523495449241695116984065, −7.04593252582326949441064573977, −5.75571183266589099114723904188, −4.56682612792375265863773380308, −3.60471181990199605185861543506, −1.73380411340543148040567092335, −0.44954490745376720150539643938,
1.22017484790844156879893549390, 3.20690432957198147676062115888, 4.93331658836806529586965035953, 5.67672860243707417820474182667, 6.44690426756993843870090331482, 7.16373096671150497095547832174, 8.275076377467336873976566002800, 9.318856381768392393059731657438, 9.843104634515465691445669145232, 11.14427184444146006606573593772
