| L(s) = 1 | + (0.0713 + 1.41i)2-s + (0.118 − 0.154i)3-s + (−1.98 + 0.201i)4-s + (2.39 − 1.83i)5-s + (0.226 + 0.156i)6-s + (1.47 − 2.19i)7-s + (−0.426 − 2.79i)8-s + (0.766 + 2.86i)9-s + (2.76 + 3.25i)10-s + (5.43 − 0.715i)11-s + (−0.204 + 0.330i)12-s + (−5.50 + 2.28i)13-s + (3.20 + 1.92i)14-s − 0.587i·15-s + (3.91 − 0.802i)16-s + (−3.70 + 2.13i)17-s + ⋯ |
| L(s) = 1 | + (0.0504 + 0.998i)2-s + (0.0683 − 0.0890i)3-s + (−0.994 + 0.100i)4-s + (1.07 − 0.822i)5-s + (0.0924 + 0.0637i)6-s + (0.556 − 0.830i)7-s + (−0.150 − 0.988i)8-s + (0.255 + 0.953i)9-s + (0.875 + 1.02i)10-s + (1.63 − 0.215i)11-s + (−0.0590 + 0.0955i)12-s + (−1.52 + 0.632i)13-s + (0.857 + 0.514i)14-s − 0.151i·15-s + (0.979 − 0.200i)16-s + (−0.897 + 0.518i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 224 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.760 - 0.649i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 224 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.760 - 0.649i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.33892 + 0.494143i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.33892 + 0.494143i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.0713 - 1.41i)T \) |
| 7 | \( 1 + (-1.47 + 2.19i)T \) |
| good | 3 | \( 1 + (-0.118 + 0.154i)T + (-0.776 - 2.89i)T^{2} \) |
| 5 | \( 1 + (-2.39 + 1.83i)T + (1.29 - 4.82i)T^{2} \) |
| 11 | \( 1 + (-5.43 + 0.715i)T + (10.6 - 2.84i)T^{2} \) |
| 13 | \( 1 + (5.50 - 2.28i)T + (9.19 - 9.19i)T^{2} \) |
| 17 | \( 1 + (3.70 - 2.13i)T + (8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-0.183 + 1.39i)T + (-18.3 - 4.91i)T^{2} \) |
| 23 | \( 1 + (0.326 + 1.21i)T + (-19.9 + 11.5i)T^{2} \) |
| 29 | \( 1 + (-0.497 - 1.20i)T + (-20.5 + 20.5i)T^{2} \) |
| 31 | \( 1 + (0.251 + 0.435i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (-5.39 + 4.14i)T + (9.57 - 35.7i)T^{2} \) |
| 41 | \( 1 + (3.02 - 3.02i)T - 41iT^{2} \) |
| 43 | \( 1 + (3.49 - 8.44i)T + (-30.4 - 30.4i)T^{2} \) |
| 47 | \( 1 + (7.99 + 4.61i)T + (23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (7.09 - 0.934i)T + (51.1 - 13.7i)T^{2} \) |
| 59 | \( 1 + (-0.00968 - 0.0735i)T + (-56.9 + 15.2i)T^{2} \) |
| 61 | \( 1 + (-9.08 - 1.19i)T + (58.9 + 15.7i)T^{2} \) |
| 67 | \( 1 + (-0.469 + 0.611i)T + (-17.3 - 64.7i)T^{2} \) |
| 71 | \( 1 + (7.84 + 7.84i)T + 71iT^{2} \) |
| 73 | \( 1 + (6.92 + 1.85i)T + (63.2 + 36.5i)T^{2} \) |
| 79 | \( 1 + (0.877 + 0.506i)T + (39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (6.18 - 2.56i)T + (58.6 - 58.6i)T^{2} \) |
| 89 | \( 1 + (13.9 - 3.74i)T + (77.0 - 44.5i)T^{2} \) |
| 97 | \( 1 - 0.451T + 97T^{2} \) |
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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
−12.80395776634062677275461669431, −11.45933233282777151737054164300, −10.02274464829052311694022695333, −9.315701146649601008151826733870, −8.379350841161661528882518063417, −7.21995854153532915988532348076, −6.34827554839283406085535377216, −4.93045947237941838547950964501, −4.36988855314605098987168118813, −1.64170668305308125072621816043,
1.85658499435729140469929953883, 3.00780711511099045571642097088, 4.53813915375055628285351238021, 5.82529264831446093337939863081, 6.93389957256011722546158455001, 8.663345619215587046796072776899, 9.631906829115814274595506272367, 9.943091763863756241606377212761, 11.38273103639728238796261951376, 11.97662889373803573912337295288
