| L(s) = 1 | + (0.365 + 0.930i)2-s + (−0.0747 − 0.997i)3-s + (−0.733 + 0.680i)4-s + (2.54 − 1.73i)5-s + (0.900 − 0.433i)6-s + (−0.146 − 2.64i)7-s + (−0.900 − 0.433i)8-s + (−0.988 + 0.149i)9-s + (2.54 + 1.73i)10-s + (−3.51 − 0.529i)11-s + (0.733 + 0.680i)12-s + (1.89 − 2.37i)13-s + (2.40 − 1.10i)14-s + (−1.91 − 2.40i)15-s + (0.0747 − 0.997i)16-s + (4.83 − 1.49i)17-s + ⋯ |
| L(s) = 1 | + (0.258 + 0.658i)2-s + (−0.0431 − 0.575i)3-s + (−0.366 + 0.340i)4-s + (1.13 − 0.775i)5-s + (0.367 − 0.177i)6-s + (−0.0552 − 0.998i)7-s + (−0.318 − 0.153i)8-s + (−0.329 + 0.0496i)9-s + (0.804 + 0.548i)10-s + (−1.05 − 0.159i)11-s + (0.211 + 0.196i)12-s + (0.525 − 0.659i)13-s + (0.642 − 0.294i)14-s + (−0.495 − 0.621i)15-s + (0.0186 − 0.249i)16-s + (1.17 − 0.361i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 294 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.893 + 0.449i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 294 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.893 + 0.449i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.52672 - 0.362739i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.52672 - 0.362739i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.365 - 0.930i)T \) |
| 3 | \( 1 + (0.0747 + 0.997i)T \) |
| 7 | \( 1 + (0.146 + 2.64i)T \) |
| good | 5 | \( 1 + (-2.54 + 1.73i)T + (1.82 - 4.65i)T^{2} \) |
| 11 | \( 1 + (3.51 + 0.529i)T + (10.5 + 3.24i)T^{2} \) |
| 13 | \( 1 + (-1.89 + 2.37i)T + (-2.89 - 12.6i)T^{2} \) |
| 17 | \( 1 + (-4.83 + 1.49i)T + (14.0 - 9.57i)T^{2} \) |
| 19 | \( 1 + (3.52 - 6.09i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-5.24 - 1.61i)T + (19.0 + 12.9i)T^{2} \) |
| 29 | \( 1 + (-0.362 + 1.58i)T + (-26.1 - 12.5i)T^{2} \) |
| 31 | \( 1 + (-4.73 - 8.20i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (-6.71 - 6.23i)T + (2.76 + 36.8i)T^{2} \) |
| 41 | \( 1 + (10.7 + 5.18i)T + (25.5 + 32.0i)T^{2} \) |
| 43 | \( 1 + (4.44 - 2.13i)T + (26.8 - 33.6i)T^{2} \) |
| 47 | \( 1 + (2.25 + 5.75i)T + (-34.4 + 31.9i)T^{2} \) |
| 53 | \( 1 + (3.68 - 3.41i)T + (3.96 - 52.8i)T^{2} \) |
| 59 | \( 1 + (-6.08 - 4.14i)T + (21.5 + 54.9i)T^{2} \) |
| 61 | \( 1 + (-5.21 - 4.84i)T + (4.55 + 60.8i)T^{2} \) |
| 67 | \( 1 + (3.03 + 5.26i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (2.51 + 11.0i)T + (-63.9 + 30.8i)T^{2} \) |
| 73 | \( 1 + (3.04 - 7.76i)T + (-53.5 - 49.6i)T^{2} \) |
| 79 | \( 1 + (2.47 - 4.28i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (-3.01 - 3.78i)T + (-18.4 + 80.9i)T^{2} \) |
| 89 | \( 1 + (-10.1 + 1.52i)T + (85.0 - 26.2i)T^{2} \) |
| 97 | \( 1 + 3.36T + 97T^{2} \) |
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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
−12.06221908382812911407823497162, −10.48848329748058762967868156682, −9.928067598990826954778874841920, −8.504377852900545063904876838878, −7.88536867977084750539721538755, −6.70948924192702267299366927665, −5.69101394435072819079451220288, −4.96994237026719530650895020384, −3.24380368206299646619228781666, −1.24902249126621408591716873432,
2.23570413457273822368320701041, 3.05632183765308853016238683500, 4.76251298739054738295890159490, 5.70971318462882502303359889216, 6.58397841968928846549216859545, 8.348497213982803654059332750698, 9.378672050867649868284788849087, 10.02998606986265873877971744906, 10.88973670534990269025735142877, 11.60389548767991052236933786611
