| L(s) = 1 | + (1.38 − 0.283i)2-s + (−0.881 − 0.471i)3-s + (1.83 − 0.785i)4-s + (−0.311 + 0.379i)5-s + (−1.35 − 0.403i)6-s + (2.05 + 1.37i)7-s + (2.32 − 1.60i)8-s + (0.555 + 0.831i)9-s + (−0.324 + 0.614i)10-s + (0.708 + 0.215i)11-s + (−1.99 − 0.174i)12-s + (1.87 − 1.54i)13-s + (3.24 + 1.32i)14-s + (0.453 − 0.187i)15-s + (2.76 − 2.88i)16-s + (0.338 + 0.140i)17-s + ⋯ |
| L(s) = 1 | + (0.979 − 0.200i)2-s + (−0.509 − 0.272i)3-s + (0.919 − 0.392i)4-s + (−0.139 + 0.169i)5-s + (−0.553 − 0.164i)6-s + (0.777 + 0.519i)7-s + (0.822 − 0.569i)8-s + (0.185 + 0.277i)9-s + (−0.102 + 0.194i)10-s + (0.213 + 0.0648i)11-s + (−0.575 − 0.0502i)12-s + (0.521 − 0.427i)13-s + (0.866 + 0.353i)14-s + (0.117 − 0.0485i)15-s + (0.691 − 0.722i)16-s + (0.0822 + 0.0340i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.876 + 0.480i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.876 + 0.480i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.18586 - 0.559941i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.18586 - 0.559941i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-1.38 + 0.283i)T \) |
| 3 | \( 1 + (0.881 + 0.471i)T \) |
| good | 5 | \( 1 + (0.311 - 0.379i)T + (-0.975 - 4.90i)T^{2} \) |
| 7 | \( 1 + (-2.05 - 1.37i)T + (2.67 + 6.46i)T^{2} \) |
| 11 | \( 1 + (-0.708 - 0.215i)T + (9.14 + 6.11i)T^{2} \) |
| 13 | \( 1 + (-1.87 + 1.54i)T + (2.53 - 12.7i)T^{2} \) |
| 17 | \( 1 + (-0.338 - 0.140i)T + (12.0 + 12.0i)T^{2} \) |
| 19 | \( 1 + (-0.634 + 6.44i)T + (-18.6 - 3.70i)T^{2} \) |
| 23 | \( 1 + (7.28 + 1.44i)T + (21.2 + 8.80i)T^{2} \) |
| 29 | \( 1 + (-0.561 - 1.84i)T + (-24.1 + 16.1i)T^{2} \) |
| 31 | \( 1 + (5.57 - 5.57i)T - 31iT^{2} \) |
| 37 | \( 1 + (-6.92 + 0.681i)T + (36.2 - 7.21i)T^{2} \) |
| 41 | \( 1 + (0.989 - 4.97i)T + (-37.8 - 15.6i)T^{2} \) |
| 43 | \( 1 + (5.69 - 3.04i)T + (23.8 - 35.7i)T^{2} \) |
| 47 | \( 1 + (1.01 - 2.44i)T + (-33.2 - 33.2i)T^{2} \) |
| 53 | \( 1 + (2.92 - 9.62i)T + (-44.0 - 29.4i)T^{2} \) |
| 59 | \( 1 + (8.85 + 7.26i)T + (11.5 + 57.8i)T^{2} \) |
| 61 | \( 1 + (-0.238 + 0.445i)T + (-33.8 - 50.7i)T^{2} \) |
| 67 | \( 1 + (-1.29 + 2.41i)T + (-37.2 - 55.7i)T^{2} \) |
| 71 | \( 1 + (-0.719 + 1.07i)T + (-27.1 - 65.5i)T^{2} \) |
| 73 | \( 1 + (0.675 - 0.451i)T + (27.9 - 67.4i)T^{2} \) |
| 79 | \( 1 + (-3.41 - 8.25i)T + (-55.8 + 55.8i)T^{2} \) |
| 83 | \( 1 + (11.2 + 1.11i)T + (81.4 + 16.1i)T^{2} \) |
| 89 | \( 1 + (9.94 - 1.97i)T + (82.2 - 34.0i)T^{2} \) |
| 97 | \( 1 + (-1.02 + 1.02i)T - 97iT^{2} \) |
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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
−11.22290589663022689618589628469, −10.97597744798518270507786180914, −9.646090024940872781561859601972, −8.285875591158980316965877695559, −7.26137764927665780460418201697, −6.27853083913904462674373850411, −5.35538030299224001002000704031, −4.47712683045955537079779590812, −3.05127339966517342619182212923, −1.60411158065659281031962977796,
1.77710229957004080504139516424, 3.77687349066173742865966072683, 4.36731414447397830681258373694, 5.59364424426620199676162935857, 6.34003483580751085176651148794, 7.59467055968497960718331787413, 8.302989762358513271469142061791, 9.885987810078444797782193560804, 10.75877595514292558268105182763, 11.64378136740116937443839286396
