| L(s) = 1 | + (1.29 − 0.577i)2-s + (−3.03 + 0.298i)3-s + (1.33 − 1.49i)4-s + (1.49 − 2.79i)5-s + (−3.74 + 2.13i)6-s + (0.0999 − 0.502i)7-s + (0.862 − 2.69i)8-s + (6.17 − 1.22i)9-s + (0.316 − 4.47i)10-s + (0.255 − 0.209i)11-s + (−3.60 + 4.92i)12-s + (−5.32 + 2.84i)13-s + (−0.160 − 0.706i)14-s + (−3.70 + 8.93i)15-s + (−0.440 − 3.97i)16-s + (2.10 + 5.08i)17-s + ⋯ |
| L(s) = 1 | + (0.912 − 0.408i)2-s + (−1.75 + 0.172i)3-s + (0.667 − 0.745i)4-s + (0.668 − 1.25i)5-s + (−1.52 + 0.872i)6-s + (0.0377 − 0.189i)7-s + (0.304 − 0.952i)8-s + (2.05 − 0.409i)9-s + (0.100 − 1.41i)10-s + (0.0770 − 0.0632i)11-s + (−1.04 + 1.42i)12-s + (−1.47 + 0.789i)13-s + (−0.0430 − 0.188i)14-s + (−0.955 + 2.30i)15-s + (−0.110 − 0.993i)16-s + (0.510 + 1.23i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 128 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.239 + 0.970i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 128 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.239 + 0.970i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.927136 - 0.725945i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.927136 - 0.725945i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-1.29 + 0.577i)T \) |
| good | 3 | \( 1 + (3.03 - 0.298i)T + (2.94 - 0.585i)T^{2} \) |
| 5 | \( 1 + (-1.49 + 2.79i)T + (-2.77 - 4.15i)T^{2} \) |
| 7 | \( 1 + (-0.0999 + 0.502i)T + (-6.46 - 2.67i)T^{2} \) |
| 11 | \( 1 + (-0.255 + 0.209i)T + (2.14 - 10.7i)T^{2} \) |
| 13 | \( 1 + (5.32 - 2.84i)T + (7.22 - 10.8i)T^{2} \) |
| 17 | \( 1 + (-2.10 - 5.08i)T + (-12.0 + 12.0i)T^{2} \) |
| 19 | \( 1 + (-4.72 - 1.43i)T + (15.7 + 10.5i)T^{2} \) |
| 23 | \( 1 + (-2.84 - 1.89i)T + (8.80 + 21.2i)T^{2} \) |
| 29 | \( 1 + (-0.778 + 0.948i)T + (-5.65 - 28.4i)T^{2} \) |
| 31 | \( 1 + (1.67 + 1.67i)T + 31iT^{2} \) |
| 37 | \( 1 + (0.903 + 2.97i)T + (-30.7 + 20.5i)T^{2} \) |
| 41 | \( 1 + (3.47 - 5.19i)T + (-15.6 - 37.8i)T^{2} \) |
| 43 | \( 1 + (0.477 + 0.0470i)T + (42.1 + 8.38i)T^{2} \) |
| 47 | \( 1 + (-0.997 + 0.412i)T + (33.2 - 33.2i)T^{2} \) |
| 53 | \( 1 + (-4.57 - 5.57i)T + (-10.3 + 51.9i)T^{2} \) |
| 59 | \( 1 + (3.56 + 1.90i)T + (32.7 + 49.0i)T^{2} \) |
| 61 | \( 1 + (0.319 + 3.23i)T + (-59.8 + 11.9i)T^{2} \) |
| 67 | \( 1 + (-0.677 - 6.87i)T + (-65.7 + 13.0i)T^{2} \) |
| 71 | \( 1 + (-1.28 - 0.254i)T + (65.5 + 27.1i)T^{2} \) |
| 73 | \( 1 + (1.26 + 6.38i)T + (-67.4 + 27.9i)T^{2} \) |
| 79 | \( 1 + (12.4 + 5.16i)T + (55.8 + 55.8i)T^{2} \) |
| 83 | \( 1 + (3.39 - 11.1i)T + (-69.0 - 46.1i)T^{2} \) |
| 89 | \( 1 + (-0.951 + 0.635i)T + (34.0 - 82.2i)T^{2} \) |
| 97 | \( 1 + (13.2 + 13.2i)T + 97iT^{2} \) |
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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
−12.70050963718818537979449910725, −12.24312927466761215486166414797, −11.40562470759866224544445465326, −10.22190446020452218923223739477, −9.509397456271393153687536177680, −7.18894680984119121691294171764, −5.88013993213166741783466627562, −5.21912615266648223641230119605, −4.32022785824439698809355345365, −1.38516080503868109346726010401,
2.80684857032284140622750117440, 5.03359729618277103100974997307, 5.59942131869022677452165671039, 6.90029493639688588463148439450, 7.27693328879352790827088206409, 9.907834839360968353954608217747, 10.75845230912573720039817077480, 11.74466169735366401088648235167, 12.34231322532206391367646809581, 13.53416306476064042052643412725
