| L(s) = 1 | + (0.173 − 0.984i)2-s + (1.67 + 0.431i)3-s + (−0.939 − 0.342i)4-s + (1.14 + 3.13i)5-s + (0.716 − 1.57i)6-s + (−1.07 − 1.85i)7-s + (−0.5 + 0.866i)8-s + (2.62 + 1.44i)9-s + (3.28 − 0.579i)10-s + (−5.41 − 3.12i)11-s + (−1.42 − 0.979i)12-s + (−2.56 − 3.05i)13-s + (−2.01 + 0.734i)14-s + (0.560 + 5.75i)15-s + (0.766 + 0.642i)16-s + (−0.403 − 0.0711i)17-s + ⋯ |
| L(s) = 1 | + (0.122 − 0.696i)2-s + (0.968 + 0.249i)3-s + (−0.469 − 0.171i)4-s + (0.510 + 1.40i)5-s + (0.292 − 0.643i)6-s + (−0.405 − 0.702i)7-s + (−0.176 + 0.306i)8-s + (0.875 + 0.482i)9-s + (1.03 − 0.183i)10-s + (−1.63 − 0.943i)11-s + (−0.412 − 0.282i)12-s + (−0.710 − 0.846i)13-s + (−0.539 + 0.196i)14-s + (0.144 + 1.48i)15-s + (0.191 + 0.160i)16-s + (−0.0978 − 0.0172i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 114 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.930 + 0.365i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 114 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.930 + 0.365i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.33342 - 0.252193i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.33342 - 0.252193i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.173 + 0.984i)T \) |
| 3 | \( 1 + (-1.67 - 0.431i)T \) |
| 19 | \( 1 + (-4.34 + 0.329i)T \) |
| good | 5 | \( 1 + (-1.14 - 3.13i)T + (-3.83 + 3.21i)T^{2} \) |
| 7 | \( 1 + (1.07 + 1.85i)T + (-3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (5.41 + 3.12i)T + (5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (2.56 + 3.05i)T + (-2.25 + 12.8i)T^{2} \) |
| 17 | \( 1 + (0.403 + 0.0711i)T + (15.9 + 5.81i)T^{2} \) |
| 23 | \( 1 + (0.280 - 0.770i)T + (-17.6 - 14.7i)T^{2} \) |
| 29 | \( 1 + (-0.805 - 4.56i)T + (-27.2 + 9.91i)T^{2} \) |
| 31 | \( 1 + (2.02 - 1.16i)T + (15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 - 6.01iT - 37T^{2} \) |
| 41 | \( 1 + (-0.926 - 0.777i)T + (7.11 + 40.3i)T^{2} \) |
| 43 | \( 1 + (-5.87 + 2.13i)T + (32.9 - 27.6i)T^{2} \) |
| 47 | \( 1 + (7.59 - 1.33i)T + (44.1 - 16.0i)T^{2} \) |
| 53 | \( 1 + (0.220 + 0.0802i)T + (40.6 + 34.0i)T^{2} \) |
| 59 | \( 1 + (-0.930 + 5.27i)T + (-55.4 - 20.1i)T^{2} \) |
| 61 | \( 1 + (-7.30 - 2.65i)T + (46.7 + 39.2i)T^{2} \) |
| 67 | \( 1 + (-3.48 + 0.614i)T + (62.9 - 22.9i)T^{2} \) |
| 71 | \( 1 + (-4.19 + 1.52i)T + (54.3 - 45.6i)T^{2} \) |
| 73 | \( 1 + (4.33 + 3.63i)T + (12.6 + 71.8i)T^{2} \) |
| 79 | \( 1 + (8.05 - 9.59i)T + (-13.7 - 77.7i)T^{2} \) |
| 83 | \( 1 + (-8.01 + 4.62i)T + (41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + (5.61 - 4.71i)T + (15.4 - 87.6i)T^{2} \) |
| 97 | \( 1 + (16.0 + 2.83i)T + (91.1 + 33.1i)T^{2} \) |
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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
−13.60592128396102487601644051938, −12.85776320182177351781182275783, −11.01163731194803310900725370081, −10.34380500809456920032566337423, −9.745670671319601744156541285195, −8.111503613329624352762278843883, −7.09578501697515224646991153925, −5.31935155958272448085992797205, −3.32136686441162803471039806262, −2.71487538198983525595867719210,
2.34059615826681349755095999868, 4.53584581617290505220326034315, 5.57881508231541731927475100820, 7.25975451281761104190035676622, 8.224617009009099915197045550490, 9.310028009968613094214829893903, 9.794302412235428772467292120253, 12.24339507368346136117161501361, 12.83435064254517825779387854396, 13.56071282024081886167599116633
