| L(s) = 1 | + (−0.415 − 0.909i)2-s + (0.735 + 0.472i)3-s + (−0.654 + 0.755i)4-s + (0.419 + 2.91i)5-s + (0.124 − 0.864i)6-s + (1.51 + 3.31i)7-s + (0.959 + 0.281i)8-s + (−0.929 − 2.03i)9-s + (2.47 − 1.59i)10-s + (−0.437 − 3.03i)11-s + (−0.838 + 0.246i)12-s + (1.01 − 0.296i)13-s + (2.38 − 2.75i)14-s + (−1.06 + 2.34i)15-s + (−0.142 − 0.989i)16-s + (−1.63 − 1.88i)17-s + ⋯ |
| L(s) = 1 | + (−0.293 − 0.643i)2-s + (0.424 + 0.272i)3-s + (−0.327 + 0.377i)4-s + (0.187 + 1.30i)5-s + (0.0507 − 0.353i)6-s + (0.572 + 1.25i)7-s + (0.339 + 0.0996i)8-s + (−0.309 − 0.678i)9-s + (0.783 − 0.503i)10-s + (−0.131 − 0.916i)11-s + (−0.242 + 0.0710i)12-s + (0.280 − 0.0822i)13-s + (0.637 − 0.736i)14-s + (−0.275 + 0.604i)15-s + (−0.0355 − 0.247i)16-s + (−0.395 − 0.456i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 134 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.965 - 0.260i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 134 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.965 - 0.260i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.07198 + 0.141984i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.07198 + 0.141984i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.415 + 0.909i)T \) |
| 67 | \( 1 + (-6.53 - 4.93i)T \) |
| good | 3 | \( 1 + (-0.735 - 0.472i)T + (1.24 + 2.72i)T^{2} \) |
| 5 | \( 1 + (-0.419 - 2.91i)T + (-4.79 + 1.40i)T^{2} \) |
| 7 | \( 1 + (-1.51 - 3.31i)T + (-4.58 + 5.29i)T^{2} \) |
| 11 | \( 1 + (0.437 + 3.03i)T + (-10.5 + 3.09i)T^{2} \) |
| 13 | \( 1 + (-1.01 + 0.296i)T + (10.9 - 7.02i)T^{2} \) |
| 17 | \( 1 + (1.63 + 1.88i)T + (-2.41 + 16.8i)T^{2} \) |
| 19 | \( 1 + (-0.843 + 1.84i)T + (-12.4 - 14.3i)T^{2} \) |
| 23 | \( 1 + (-2.48 - 1.59i)T + (9.55 + 20.9i)T^{2} \) |
| 29 | \( 1 + 2.05T + 29T^{2} \) |
| 31 | \( 1 + (-3.61 - 1.06i)T + (26.0 + 16.7i)T^{2} \) |
| 37 | \( 1 + 7.68T + 37T^{2} \) |
| 41 | \( 1 + (8.11 + 9.36i)T + (-5.83 + 40.5i)T^{2} \) |
| 43 | \( 1 + (-7.87 - 9.08i)T + (-6.11 + 42.5i)T^{2} \) |
| 47 | \( 1 + (8.52 + 5.47i)T + (19.5 + 42.7i)T^{2} \) |
| 53 | \( 1 + (-7.45 + 8.60i)T + (-7.54 - 52.4i)T^{2} \) |
| 59 | \( 1 + (-7.35 - 2.16i)T + (49.6 + 31.8i)T^{2} \) |
| 61 | \( 1 + (0.830 - 5.77i)T + (-58.5 - 17.1i)T^{2} \) |
| 71 | \( 1 + (6.65 - 7.67i)T + (-10.1 - 70.2i)T^{2} \) |
| 73 | \( 1 + (-1.65 + 11.5i)T + (-70.0 - 20.5i)T^{2} \) |
| 79 | \( 1 + (0.336 - 0.0987i)T + (66.4 - 42.7i)T^{2} \) |
| 83 | \( 1 + (1.44 + 10.0i)T + (-79.6 + 23.3i)T^{2} \) |
| 89 | \( 1 + (11.2 - 7.22i)T + (36.9 - 80.9i)T^{2} \) |
| 97 | \( 1 + 6.12T + 97T^{2} \) |
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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
−13.37795579758438097999476345261, −11.84824656445813956614349151007, −11.32786015097344080585976087032, −10.27734480361118532651134008300, −9.072354922313789039511846916283, −8.424032031435940869677219773818, −6.80949583030613189468597777680, −5.46942014271703409863074756388, −3.42039126700572343056556557374, −2.52221328658432907975254656145,
1.53049903259449827187188354598, 4.33605242595149922034350468409, 5.24650149138098488522274355964, 6.98569711461055427034964799369, 7.991182647670051202028841289268, 8.680612412571284106023964103142, 9.921336091454425983041209144311, 10.97581477063143903335631573031, 12.53233963729939633121501861050, 13.42439596191590827945225966182
