| L(s) = 1 | + (1.27 − 0.615i)2-s + (0.731 − 0.391i)3-s + (1.24 − 1.56i)4-s + (−0.0771 − 0.0939i)5-s + (0.690 − 0.948i)6-s + (−3.30 + 2.21i)7-s + (0.616 − 2.76i)8-s + (−1.28 + 1.92i)9-s + (−0.156 − 0.0721i)10-s + (0.0222 − 0.00674i)11-s + (0.295 − 1.63i)12-s + (0.466 + 0.382i)13-s + (−2.85 + 4.85i)14-s + (−0.0931 − 0.0386i)15-s + (−0.915 − 3.89i)16-s + (4.59 − 1.90i)17-s + ⋯ |
| L(s) = 1 | + (0.900 − 0.435i)2-s + (0.422 − 0.225i)3-s + (0.620 − 0.783i)4-s + (−0.0344 − 0.0420i)5-s + (0.282 − 0.387i)6-s + (−1.25 + 0.835i)7-s + (0.217 − 0.975i)8-s + (−0.428 + 0.640i)9-s + (−0.0493 − 0.0228i)10-s + (0.00670 − 0.00203i)11-s + (0.0853 − 0.471i)12-s + (0.129 + 0.106i)13-s + (−0.761 + 1.29i)14-s + (−0.0240 − 0.00996i)15-s + (−0.228 − 0.973i)16-s + (1.11 − 0.461i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 128 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.748 + 0.663i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 128 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.748 + 0.663i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.61154 - 0.611205i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.61154 - 0.611205i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-1.27 + 0.615i)T \) |
| good | 3 | \( 1 + (-0.731 + 0.391i)T + (1.66 - 2.49i)T^{2} \) |
| 5 | \( 1 + (0.0771 + 0.0939i)T + (-0.975 + 4.90i)T^{2} \) |
| 7 | \( 1 + (3.30 - 2.21i)T + (2.67 - 6.46i)T^{2} \) |
| 11 | \( 1 + (-0.0222 + 0.00674i)T + (9.14 - 6.11i)T^{2} \) |
| 13 | \( 1 + (-0.466 - 0.382i)T + (2.53 + 12.7i)T^{2} \) |
| 17 | \( 1 + (-4.59 + 1.90i)T + (12.0 - 12.0i)T^{2} \) |
| 19 | \( 1 + (-0.166 - 1.68i)T + (-18.6 + 3.70i)T^{2} \) |
| 23 | \( 1 + (3.40 - 0.678i)T + (21.2 - 8.80i)T^{2} \) |
| 29 | \( 1 + (1.44 - 4.77i)T + (-24.1 - 16.1i)T^{2} \) |
| 31 | \( 1 + (6.39 + 6.39i)T + 31iT^{2} \) |
| 37 | \( 1 + (-8.75 - 0.862i)T + (36.2 + 7.21i)T^{2} \) |
| 41 | \( 1 + (-0.0606 - 0.304i)T + (-37.8 + 15.6i)T^{2} \) |
| 43 | \( 1 + (4.91 + 2.62i)T + (23.8 + 35.7i)T^{2} \) |
| 47 | \( 1 + (-0.167 - 0.404i)T + (-33.2 + 33.2i)T^{2} \) |
| 53 | \( 1 + (-1.85 - 6.11i)T + (-44.0 + 29.4i)T^{2} \) |
| 59 | \( 1 + (-7.95 + 6.53i)T + (11.5 - 57.8i)T^{2} \) |
| 61 | \( 1 + (6.59 + 12.3i)T + (-33.8 + 50.7i)T^{2} \) |
| 67 | \( 1 + (-5.13 - 9.61i)T + (-37.2 + 55.7i)T^{2} \) |
| 71 | \( 1 + (-3.99 - 5.97i)T + (-27.1 + 65.5i)T^{2} \) |
| 73 | \( 1 + (8.64 + 5.77i)T + (27.9 + 67.4i)T^{2} \) |
| 79 | \( 1 + (0.504 - 1.21i)T + (-55.8 - 55.8i)T^{2} \) |
| 83 | \( 1 + (-16.2 + 1.60i)T + (81.4 - 16.1i)T^{2} \) |
| 89 | \( 1 + (-2.00 - 0.399i)T + (82.2 + 34.0i)T^{2} \) |
| 97 | \( 1 + (-5.29 - 5.29i)T + 97iT^{2} \) |
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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
−13.12664386365801351833490287020, −12.42592783173172499340131471812, −11.47003241172337298270498463602, −10.16640498749729632262004349420, −9.236036434689379709960126414301, −7.74051303197713005838264093780, −6.29494077757097813518228245532, −5.35916908986838227605606513465, −3.54361039478685034745312834388, −2.40186077640909097961694043157,
3.15063414276425626669239177545, 3.92976520577926262477534952180, 5.73237807493796056113368853402, 6.75041240836745989078880534398, 7.86694831728147768943671994024, 9.243951558298754418232616425451, 10.36592087790379474279728326530, 11.68798875664846003438963447403, 12.75362570872870529964931481963, 13.48199478844117971504661932784
