| L(s) = 1 | + (0.669 − 0.743i)2-s + (−0.360 − 1.10i)3-s + (−0.104 − 0.994i)4-s + (−1.99 − 0.887i)5-s + (−1.06 − 0.474i)6-s + (1.93 − 0.410i)7-s + (−0.809 − 0.587i)8-s + (1.32 − 0.964i)9-s + (−1.99 + 0.887i)10-s − 1.72·11-s + (−1.06 + 0.474i)12-s + (2.15 + 3.73i)13-s + (0.986 − 1.70i)14-s + (−0.265 + 2.53i)15-s + (−0.978 + 0.207i)16-s + (0.434 + 4.13i)17-s + ⋯ |
| L(s) = 1 | + (0.473 − 0.525i)2-s + (−0.207 − 0.640i)3-s + (−0.0522 − 0.497i)4-s + (−0.891 − 0.396i)5-s + (−0.434 − 0.193i)6-s + (0.729 − 0.155i)7-s + (−0.286 − 0.207i)8-s + (0.442 − 0.321i)9-s + (−0.630 + 0.280i)10-s − 0.521·11-s + (−0.307 + 0.136i)12-s + (0.597 + 1.03i)13-s + (0.263 − 0.456i)14-s + (−0.0686 + 0.653i)15-s + (−0.244 + 0.0519i)16-s + (0.105 + 1.00i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 122 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0719 + 0.997i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 122 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0719 + 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.805055 - 0.865241i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.805055 - 0.865241i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.669 + 0.743i)T \) |
| 61 | \( 1 + (-6.75 - 3.92i)T \) |
| good | 3 | \( 1 + (0.360 + 1.10i)T + (-2.42 + 1.76i)T^{2} \) |
| 5 | \( 1 + (1.99 + 0.887i)T + (3.34 + 3.71i)T^{2} \) |
| 7 | \( 1 + (-1.93 + 0.410i)T + (6.39 - 2.84i)T^{2} \) |
| 11 | \( 1 + 1.72T + 11T^{2} \) |
| 13 | \( 1 + (-2.15 - 3.73i)T + (-6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 + (-0.434 - 4.13i)T + (-16.6 + 3.53i)T^{2} \) |
| 19 | \( 1 + (-3.27 - 0.696i)T + (17.3 + 7.72i)T^{2} \) |
| 23 | \( 1 + (-3.52 + 2.56i)T + (7.10 - 21.8i)T^{2} \) |
| 29 | \( 1 + (2.23 - 3.87i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (-1.85 - 2.05i)T + (-3.24 + 30.8i)T^{2} \) |
| 37 | \( 1 + (2.17 - 6.68i)T + (-29.9 - 21.7i)T^{2} \) |
| 41 | \( 1 + (1.29 - 3.99i)T + (-33.1 - 24.0i)T^{2} \) |
| 43 | \( 1 + (-1.20 + 11.4i)T + (-42.0 - 8.94i)T^{2} \) |
| 47 | \( 1 + (-1.70 + 2.95i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (8.82 + 6.40i)T + (16.3 + 50.4i)T^{2} \) |
| 59 | \( 1 + (6.44 - 7.16i)T + (-6.16 - 58.6i)T^{2} \) |
| 67 | \( 1 + (12.1 + 5.40i)T + (44.8 + 49.7i)T^{2} \) |
| 71 | \( 1 + (-6.24 + 2.78i)T + (47.5 - 52.7i)T^{2} \) |
| 73 | \( 1 + (-2.87 + 1.28i)T + (48.8 - 54.2i)T^{2} \) |
| 79 | \( 1 + (1.10 - 10.5i)T + (-77.2 - 16.4i)T^{2} \) |
| 83 | \( 1 + (-2.90 + 3.22i)T + (-8.67 - 82.5i)T^{2} \) |
| 89 | \( 1 + (3.23 + 9.94i)T + (-72.0 + 52.3i)T^{2} \) |
| 97 | \( 1 + (-8.02 - 8.90i)T + (-10.1 + 96.4i)T^{2} \) |
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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
−12.98595249179413987071979281305, −12.12186843272529608122805856203, −11.46035532290252737632144583681, −10.37559710467801760766802236899, −8.815904844731037531695498035237, −7.71293955932650652001725200773, −6.50484887339720367688238699904, −4.90466156875901440812327565338, −3.76059312226941446626422158391, −1.47082209327309720509237394893,
3.26724818913517278518797241890, 4.62857104599782135298929705152, 5.57467205343862632618812920103, 7.41042354423338594897513487138, 7.940375673696457567597541285718, 9.519149435123150265785449962189, 10.92660333741945801900210574172, 11.46137211820655734422824419989, 12.81038076592223172158086979400, 13.81597209057644475407030039671
