L(s) = 1 | + (−0.929 − 1.60i)2-s + (1.14 + 1.98i)3-s + (−0.726 + 1.25i)4-s + 0.197·5-s + (2.13 − 3.69i)6-s − 1.01·8-s + (−1.13 + 1.95i)9-s + (−0.183 − 0.317i)10-s + (2.09 + 3.62i)11-s − 3.33·12-s + (2.72 − 2.36i)13-s + (0.226 + 0.392i)15-s + (2.39 + 4.15i)16-s + (0.420 − 0.728i)17-s + 4.20·18-s + (0.675 − 1.17i)19-s + ⋯ |
L(s) = 1 | + (−0.656 − 1.13i)2-s + (0.662 + 1.14i)3-s + (−0.363 + 0.629i)4-s + 0.0882·5-s + (0.870 − 1.50i)6-s − 0.359·8-s + (−0.377 + 0.653i)9-s + (−0.0579 − 0.100i)10-s + (0.630 + 1.09i)11-s − 0.962·12-s + (0.755 − 0.655i)13-s + (0.0584 + 0.101i)15-s + (0.599 + 1.03i)16-s + (0.102 − 0.176i)17-s + 0.991·18-s + (0.155 − 0.268i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.997 + 0.0638i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.997 + 0.0638i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.37580 - 0.0439388i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.37580 - 0.0439388i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 7 | \( 1 \) |
| 13 | \( 1 + (-2.72 + 2.36i)T \) |
good | 2 | \( 1 + (0.929 + 1.60i)T + (-1 + 1.73i)T^{2} \) |
| 3 | \( 1 + (-1.14 - 1.98i)T + (-1.5 + 2.59i)T^{2} \) |
| 5 | \( 1 - 0.197T + 5T^{2} \) |
| 11 | \( 1 + (-2.09 - 3.62i)T + (-5.5 + 9.52i)T^{2} \) |
| 17 | \( 1 + (-0.420 + 0.728i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-0.675 + 1.17i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-2.05 - 3.56i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (-4.11 - 7.13i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 - 1.28T + 31T^{2} \) |
| 37 | \( 1 + (1.52 + 2.63i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (-2.69 - 4.67i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (2.66 - 4.61i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 - 11.6T + 47T^{2} \) |
| 53 | \( 1 - 4.64T + 53T^{2} \) |
| 59 | \( 1 + (-3.02 + 5.24i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (5.68 - 9.84i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (6.69 + 11.6i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (-2.98 + 5.17i)T + (-35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + 3.88T + 73T^{2} \) |
| 79 | \( 1 + 10.7T + 79T^{2} \) |
| 83 | \( 1 + 3.07T + 83T^{2} \) |
| 89 | \( 1 + (5.99 + 10.3i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (-9.73 + 16.8i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−10.35409853258190848579710919347, −9.801237476554251327158652643992, −9.136916381073565920937467487998, −8.546650762078543027471304939837, −7.29620266302173598506272044360, −5.92153523374149835178370391351, −4.61558111636670785378982816865, −3.62808008063531525655954514153, −2.82920376722463379904109306164, −1.41917180428381505523179430213,
1.01612972670841535045981335641, 2.56424057468339963261823210863, 3.90545900501267505178910851801, 5.75604521738919220624882894931, 6.39939793179825792840738426042, 7.10715899093676316488470240545, 8.013007676656682963468453449462, 8.580243355910610122703030696015, 9.135929322257562452738433101805, 10.35140578254042912618758237711