| L(s) = 1 | + (−1.31 + 0.524i)2-s + (−1.57 + 0.394i)3-s + (1.45 − 1.37i)4-s + (3.41 + 1.61i)5-s + (1.86 − 1.34i)6-s + (−1.44 − 4.74i)7-s + (−1.18 + 2.56i)8-s + (−0.320 + 0.171i)9-s + (−5.33 − 0.331i)10-s + (0.632 − 0.0938i)11-s + (−1.74 + 2.74i)12-s + (1.67 − 4.68i)13-s + (4.38 + 5.48i)14-s + (−6.01 − 1.19i)15-s + (0.207 − 3.99i)16-s + (2.48 − 0.494i)17-s + ⋯ |
| L(s) = 1 | + (−0.928 + 0.370i)2-s + (−0.909 + 0.227i)3-s + (0.725 − 0.688i)4-s + (1.52 + 0.722i)5-s + (0.760 − 0.548i)6-s + (−0.544 − 1.79i)7-s + (−0.418 + 0.908i)8-s + (−0.106 + 0.0570i)9-s + (−1.68 − 0.104i)10-s + (0.190 − 0.0282i)11-s + (−0.502 + 0.791i)12-s + (0.464 − 1.29i)13-s + (1.17 + 1.46i)14-s + (−1.55 − 0.309i)15-s + (0.0518 − 0.998i)16-s + (0.602 − 0.119i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 256 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.996 + 0.0791i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 256 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.996 + 0.0791i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.761566 - 0.0301938i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.761566 - 0.0301938i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.31 - 0.524i)T \) |
| good | 3 | \( 1 + (1.57 - 0.394i)T + (2.64 - 1.41i)T^{2} \) |
| 5 | \( 1 + (-3.41 - 1.61i)T + (3.17 + 3.86i)T^{2} \) |
| 7 | \( 1 + (1.44 + 4.74i)T + (-5.82 + 3.88i)T^{2} \) |
| 11 | \( 1 + (-0.632 + 0.0938i)T + (10.5 - 3.19i)T^{2} \) |
| 13 | \( 1 + (-1.67 + 4.68i)T + (-10.0 - 8.24i)T^{2} \) |
| 17 | \( 1 + (-2.48 + 0.494i)T + (15.7 - 6.50i)T^{2} \) |
| 19 | \( 1 + (-3.97 - 3.60i)T + (1.86 + 18.9i)T^{2} \) |
| 23 | \( 1 + (-3.51 + 0.345i)T + (22.5 - 4.48i)T^{2} \) |
| 29 | \( 1 + (2.46 + 3.32i)T + (-8.41 + 27.7i)T^{2} \) |
| 31 | \( 1 + (-2.18 - 0.904i)T + (21.9 + 21.9i)T^{2} \) |
| 37 | \( 1 + (0.0957 - 1.94i)T + (-36.8 - 3.62i)T^{2} \) |
| 41 | \( 1 + (-0.597 + 0.727i)T + (-7.99 - 40.2i)T^{2} \) |
| 43 | \( 1 + (0.415 - 1.65i)T + (-37.9 - 20.2i)T^{2} \) |
| 47 | \( 1 + (3.19 + 2.13i)T + (17.9 + 43.4i)T^{2} \) |
| 53 | \( 1 + (-1.82 + 2.46i)T + (-15.3 - 50.7i)T^{2} \) |
| 59 | \( 1 + (-3.03 - 8.47i)T + (-45.6 + 37.4i)T^{2} \) |
| 61 | \( 1 + (-7.14 + 11.9i)T + (-28.7 - 53.7i)T^{2} \) |
| 67 | \( 1 + (10.5 + 6.33i)T + (31.5 + 59.0i)T^{2} \) |
| 71 | \( 1 + (4.30 - 8.05i)T + (-39.4 - 59.0i)T^{2} \) |
| 73 | \( 1 + (0.690 - 2.27i)T + (-60.6 - 40.5i)T^{2} \) |
| 79 | \( 1 + (-1.33 - 1.99i)T + (-30.2 + 72.9i)T^{2} \) |
| 83 | \( 1 + (0.327 + 6.66i)T + (-82.6 + 8.13i)T^{2} \) |
| 89 | \( 1 + (6.05 + 0.596i)T + (87.2 + 17.3i)T^{2} \) |
| 97 | \( 1 + (-1.93 + 4.68i)T + (-68.5 - 68.5i)T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−11.49829676756972617998539335996, −10.47564724856527169395533484984, −10.33047668920080171921379769329, −9.572951112904112253853800505431, −7.922341669832243963719494031953, −6.90554153083609327778080863653, −6.08432658514708622042036557230, −5.34403574914464934057320141124, −3.13459161428272745142638335155, −1.03965600945712618949055873681,
1.47976035087937597103525885573, 2.77918907159856640938245502936, 5.28494044554280631185649209655, 6.01320928374419369106417254726, 6.79244342466377716142521682575, 8.810386044501404967006333769574, 9.091954516966887717563094662452, 9.844803055214185758816715230520, 11.22761405274807389270269359884, 11.93477200476095410939516315171
