L(s) = 1 | + (0.559 + 1.29i)2-s + (−0.224 + 0.839i)3-s + (−1.37 + 1.45i)4-s + (−0.847 − 3.16i)5-s + (−1.21 + 0.177i)6-s + (−2.65 − 0.968i)8-s + (1.94 + 1.12i)9-s + (3.63 − 2.87i)10-s + (2.87 + 0.769i)11-s + (−0.911 − 1.47i)12-s + (3.63 + 3.63i)13-s + 2.84·15-s + (−0.229 − 3.99i)16-s + (1.81 + 3.14i)17-s + (−0.369 + 3.15i)18-s + (−1.60 + 0.429i)19-s + ⋯ |
L(s) = 1 | + (0.395 + 0.918i)2-s + (−0.129 + 0.484i)3-s + (−0.686 + 0.727i)4-s + (−0.379 − 1.41i)5-s + (−0.496 + 0.0726i)6-s + (−0.939 − 0.342i)8-s + (0.648 + 0.374i)9-s + (1.14 − 0.908i)10-s + (0.865 + 0.231i)11-s + (−0.263 − 0.426i)12-s + (1.00 + 1.00i)13-s + 0.734·15-s + (−0.0574 − 0.998i)16-s + (0.441 + 0.763i)17-s + (−0.0870 + 0.743i)18-s + (−0.367 + 0.0985i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.317 - 0.948i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.317 - 0.948i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.972560 + 1.35180i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.972560 + 1.35180i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.559 - 1.29i)T \) |
| 7 | \( 1 \) |
good | 3 | \( 1 + (0.224 - 0.839i)T + (-2.59 - 1.5i)T^{2} \) |
| 5 | \( 1 + (0.847 + 3.16i)T + (-4.33 + 2.5i)T^{2} \) |
| 11 | \( 1 + (-2.87 - 0.769i)T + (9.52 + 5.5i)T^{2} \) |
| 13 | \( 1 + (-3.63 - 3.63i)T + 13iT^{2} \) |
| 17 | \( 1 + (-1.81 - 3.14i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (1.60 - 0.429i)T + (16.4 - 9.5i)T^{2} \) |
| 23 | \( 1 + (5.33 + 3.08i)T + (11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (-5.10 - 5.10i)T + 29iT^{2} \) |
| 31 | \( 1 + (-1.00 - 1.74i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (-1.49 - 5.57i)T + (-32.0 + 18.5i)T^{2} \) |
| 41 | \( 1 - 3.71iT - 41T^{2} \) |
| 43 | \( 1 + (2.91 - 2.91i)T - 43iT^{2} \) |
| 47 | \( 1 + (-5.06 + 8.77i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (-3.68 - 0.986i)T + (45.8 + 26.5i)T^{2} \) |
| 59 | \( 1 + (-3.64 - 0.977i)T + (51.0 + 29.5i)T^{2} \) |
| 61 | \( 1 + (-6.54 + 1.75i)T + (52.8 - 30.5i)T^{2} \) |
| 67 | \( 1 + (1.57 - 5.88i)T + (-58.0 - 33.5i)T^{2} \) |
| 71 | \( 1 + 9.55iT - 71T^{2} \) |
| 73 | \( 1 + (-0.989 + 0.571i)T + (36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-0.120 + 0.209i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (-0.459 - 0.459i)T + 83iT^{2} \) |
| 89 | \( 1 + (3.76 + 2.17i)T + (44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + 6.80T + 97T^{2} \) |
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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
−10.34684292281801617263475932614, −9.448665204129764572422977413084, −8.565120122859064015160475806619, −8.256402667879777503507972622087, −6.94392434163437549690033888972, −6.14495256596787900354767069504, −5.02742501510861127690676562901, −4.28703139318061908761939548065, −3.82989813256521388043584710756, −1.41038772674305009362128177627,
0.892752394078634072700796779691, 2.39919438812675828423704254369, 3.50520975201584675791737846311, 4.09298899149375767116712686672, 5.77127391755548899918229270191, 6.38715109005921436538363428982, 7.33668034620979710493859131353, 8.325830784058353768279518670576, 9.553726331854407741840635186033, 10.23531010103336011297854987840