| L(s) = 1 | + (−0.841 + 1.13i)2-s + (1.33 + 0.524i)3-s + (−0.584 − 1.91i)4-s + (−1.33 + 0.200i)5-s + (−1.72 + 1.07i)6-s + (0.0948 − 2.64i)7-s + (2.66 + 0.945i)8-s + (−0.689 − 0.639i)9-s + (0.892 − 1.68i)10-s + (4.38 − 4.06i)11-s + (0.222 − 2.86i)12-s + (0.410 + 1.80i)13-s + (2.92 + 2.33i)14-s + (−1.88 − 0.429i)15-s + (−3.31 + 2.23i)16-s + (4.98 − 0.373i)17-s + ⋯ |
| L(s) = 1 | + (−0.594 + 0.803i)2-s + (0.771 + 0.302i)3-s + (−0.292 − 0.956i)4-s + (−0.595 + 0.0897i)5-s + (−0.702 + 0.439i)6-s + (0.0358 − 0.999i)7-s + (0.942 + 0.334i)8-s + (−0.229 − 0.213i)9-s + (0.282 − 0.531i)10-s + (1.32 − 1.22i)11-s + (0.0642 − 0.826i)12-s + (0.113 + 0.499i)13-s + (0.781 + 0.623i)14-s + (−0.486 − 0.111i)15-s + (−0.829 + 0.558i)16-s + (1.20 − 0.0906i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 392 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.988 - 0.153i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 392 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.988 - 0.153i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.18846 + 0.0915537i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.18846 + 0.0915537i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.841 - 1.13i)T \) |
| 7 | \( 1 + (-0.0948 + 2.64i)T \) |
| good | 3 | \( 1 + (-1.33 - 0.524i)T + (2.19 + 2.04i)T^{2} \) |
| 5 | \( 1 + (1.33 - 0.200i)T + (4.77 - 1.47i)T^{2} \) |
| 11 | \( 1 + (-4.38 + 4.06i)T + (0.822 - 10.9i)T^{2} \) |
| 13 | \( 1 + (-0.410 - 1.80i)T + (-11.7 + 5.64i)T^{2} \) |
| 17 | \( 1 + (-4.98 + 0.373i)T + (16.8 - 2.53i)T^{2} \) |
| 19 | \( 1 + (-3.75 - 2.16i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (2.60 + 0.195i)T + (22.7 + 3.42i)T^{2} \) |
| 29 | \( 1 + (-4.05 + 8.41i)T + (-18.0 - 22.6i)T^{2} \) |
| 31 | \( 1 + (-1.75 - 3.03i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (-0.211 - 0.310i)T + (-13.5 + 34.4i)T^{2} \) |
| 41 | \( 1 + (1.39 - 1.11i)T + (9.12 - 39.9i)T^{2} \) |
| 43 | \( 1 + (4.91 - 6.16i)T + (-9.56 - 41.9i)T^{2} \) |
| 47 | \( 1 + (-4.83 - 1.49i)T + (38.8 + 26.4i)T^{2} \) |
| 53 | \( 1 + (1.35 - 1.99i)T + (-19.3 - 49.3i)T^{2} \) |
| 59 | \( 1 + (-0.602 + 3.99i)T + (-56.3 - 17.3i)T^{2} \) |
| 61 | \( 1 + (-7.73 + 5.27i)T + (22.2 - 56.7i)T^{2} \) |
| 67 | \( 1 + (-1.76 - 3.06i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (-2.76 - 5.73i)T + (-44.2 + 55.5i)T^{2} \) |
| 73 | \( 1 + (1.69 + 5.49i)T + (-60.3 + 41.1i)T^{2} \) |
| 79 | \( 1 + (-4.77 - 2.75i)T + (39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (7.28 + 1.66i)T + (74.7 + 36.0i)T^{2} \) |
| 89 | \( 1 + (11.1 - 11.9i)T + (-6.65 - 88.7i)T^{2} \) |
| 97 | \( 1 - 0.206iT - 97T^{2} \) |
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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
−11.29498839636060953849193638027, −9.968652085460286006111609170952, −9.485793419411720234723856987424, −8.302999976649754745497251060902, −7.931853432582877177205236152557, −6.73297104718388192252939643907, −5.84619481710657653763572653892, −4.18317578526004427709093340011, −3.44981199778200260226826984420, −1.02758089234791819611564496245,
1.61564363670560573553003730508, 2.84531856898032211494596467066, 3.85980473614363247336434311150, 5.27281777076549698625467893858, 7.01909709380640114191189588870, 7.87704010819835136007015471711, 8.616106623751035286228123244888, 9.368881751935693738571403836234, 10.19227636634322055470565460574, 11.56131321923920386177159411226
