| L(s) = 1 | + (0.900 − 0.433i)2-s + (−1.04 − 1.38i)3-s + (0.623 − 0.781i)4-s + (0.438 + 0.406i)5-s + (−1.54 − 0.791i)6-s + (−1.22 + 2.34i)7-s + (0.222 − 0.974i)8-s + (−0.816 + 2.88i)9-s + (0.571 + 0.176i)10-s + (4.19 + 2.85i)11-s + (−1.73 − 0.0444i)12-s + (2.55 + 1.74i)13-s + (−0.0846 + 2.64i)14-s + (0.103 − 1.03i)15-s + (−0.222 − 0.974i)16-s + (−0.957 + 0.144i)17-s + ⋯ |
| L(s) = 1 | + (0.637 − 0.306i)2-s + (−0.603 − 0.797i)3-s + (0.311 − 0.390i)4-s + (0.196 + 0.181i)5-s + (−0.628 − 0.323i)6-s + (−0.462 + 0.886i)7-s + (0.0786 − 0.344i)8-s + (−0.272 + 0.962i)9-s + (0.180 + 0.0557i)10-s + (1.26 + 0.862i)11-s + (−0.499 − 0.0128i)12-s + (0.709 + 0.483i)13-s + (−0.0226 + 0.706i)14-s + (0.0268 − 0.266i)15-s + (−0.0556 − 0.243i)16-s + (−0.232 + 0.0350i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.991 + 0.129i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.991 + 0.129i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.93192 - 0.125366i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.93192 - 0.125366i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.900 + 0.433i)T \) |
| 3 | \( 1 + (1.04 + 1.38i)T \) |
| 7 | \( 1 + (1.22 - 2.34i)T \) |
| good | 5 | \( 1 + (-0.438 - 0.406i)T + (0.373 + 4.98i)T^{2} \) |
| 11 | \( 1 + (-4.19 - 2.85i)T + (4.01 + 10.2i)T^{2} \) |
| 13 | \( 1 + (-2.55 - 1.74i)T + (4.74 + 12.1i)T^{2} \) |
| 17 | \( 1 + (0.957 - 0.144i)T + (16.2 - 5.01i)T^{2} \) |
| 19 | \( 1 + (-1.89 - 3.28i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-0.500 - 1.27i)T + (-16.8 + 15.6i)T^{2} \) |
| 29 | \( 1 + (-5.33 + 0.803i)T + (27.7 - 8.54i)T^{2} \) |
| 31 | \( 1 + 4.97T + 31T^{2} \) |
| 37 | \( 1 + (-0.0131 + 0.0334i)T + (-27.1 - 25.1i)T^{2} \) |
| 41 | \( 1 + (6.06 - 1.87i)T + (33.8 - 23.0i)T^{2} \) |
| 43 | \( 1 + (-6.72 - 2.07i)T + (35.5 + 24.2i)T^{2} \) |
| 47 | \( 1 + (-2.40 + 1.15i)T + (29.3 - 36.7i)T^{2} \) |
| 53 | \( 1 + (-1.91 - 4.89i)T + (-38.8 + 36.0i)T^{2} \) |
| 59 | \( 1 + (1.12 + 4.94i)T + (-53.1 + 25.5i)T^{2} \) |
| 61 | \( 1 + (4.89 + 6.13i)T + (-13.5 + 59.4i)T^{2} \) |
| 67 | \( 1 - 14.8T + 67T^{2} \) |
| 71 | \( 1 + (7.34 - 9.21i)T + (-15.7 - 69.2i)T^{2} \) |
| 73 | \( 1 + (5.86 - 4.00i)T + (26.6 - 67.9i)T^{2} \) |
| 79 | \( 1 - 6.25T + 79T^{2} \) |
| 83 | \( 1 + (-1.05 + 0.719i)T + (30.3 - 77.2i)T^{2} \) |
| 89 | \( 1 + (-0.133 - 1.78i)T + (-88.0 + 13.2i)T^{2} \) |
| 97 | \( 1 + (-4.87 + 8.44i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−10.23881486085048087405727721395, −9.395545364404091012656857929735, −8.455187098273141173813818875869, −7.20875641411773328538496149509, −6.40525350474124879761422238563, −5.98456846683648006754280296585, −4.87646676210976065137627519798, −3.75624982791315302392210149772, −2.39693105671768329358868310275, −1.44191723574438144645567498587,
0.933626198078001592919515851823, 3.26286131046298338323572104910, 3.83256946465000174664695912994, 4.81665282340252374017284732683, 5.80411734264069819960397967737, 6.47262895394217480033400977705, 7.27543824481660484935837528167, 8.706073909706355158482374224382, 9.249234445942926762118957106284, 10.34263916974194985057812101767
