| L(s) = 1 | + (−0.870 + 0.492i)2-s + (−0.777 − 0.729i)3-s + (0.515 − 0.856i)4-s + (2.53 + 0.0931i)5-s + (1.03 + 0.252i)6-s + (0.00860 + 0.0440i)7-s + (−0.0275 + 0.999i)8-s + (−0.119 − 1.85i)9-s + (−2.25 + 1.16i)10-s + (−2.31 − 3.14i)11-s + (−1.02 + 0.290i)12-s + (−6.55 − 2.80i)13-s + (−0.0291 − 0.0341i)14-s + (−1.90 − 1.92i)15-s + (−0.467 − 0.883i)16-s + (7.46 − 0.137i)17-s + ⋯ |
| L(s) = 1 | + (−0.615 + 0.347i)2-s + (−0.448 − 0.421i)3-s + (0.257 − 0.428i)4-s + (1.13 + 0.0416i)5-s + (0.422 + 0.102i)6-s + (0.00325 + 0.0166i)7-s + (−0.00974 + 0.353i)8-s + (−0.0399 − 0.619i)9-s + (−0.712 + 0.368i)10-s + (−0.697 − 0.949i)11-s + (−0.296 + 0.0837i)12-s + (−1.81 − 0.778i)13-s + (−0.00779 − 0.00911i)14-s + (−0.491 − 0.495i)15-s + (−0.116 − 0.220i)16-s + (1.81 − 0.0332i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 722 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 722 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.373445 - 0.604251i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.373445 - 0.604251i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.870 - 0.492i)T \) |
| 19 | \( 1 + (3.78 - 2.16i)T \) |
| good | 3 | \( 1 + (0.777 + 0.729i)T + (0.192 + 2.99i)T^{2} \) |
| 5 | \( 1 + (-2.53 - 0.0931i)T + (4.98 + 0.367i)T^{2} \) |
| 7 | \( 1 + (-0.00860 - 0.0440i)T + (-6.48 + 2.63i)T^{2} \) |
| 11 | \( 1 + (2.31 + 3.14i)T + (-3.28 + 10.4i)T^{2} \) |
| 13 | \( 1 + (6.55 + 2.80i)T + (8.97 + 9.40i)T^{2} \) |
| 17 | \( 1 + (-7.46 + 0.137i)T + (16.9 - 0.624i)T^{2} \) |
| 23 | \( 1 + (2.54 + 0.770i)T + (19.1 + 12.7i)T^{2} \) |
| 29 | \( 1 + (1.98 + 1.60i)T + (6.08 + 28.3i)T^{2} \) |
| 31 | \( 1 + (-3.08 - 8.25i)T + (-23.3 + 20.3i)T^{2} \) |
| 37 | \( 1 + (-3.09 + 7.05i)T + (-25.0 - 27.2i)T^{2} \) |
| 41 | \( 1 + (8.50 + 0.783i)T + (40.3 + 7.49i)T^{2} \) |
| 43 | \( 1 + (0.547 + 11.9i)T + (-42.8 + 3.94i)T^{2} \) |
| 47 | \( 1 + (5.04 - 5.28i)T + (-2.15 - 46.9i)T^{2} \) |
| 53 | \( 1 + (2.34 + 8.01i)T + (-44.6 + 28.5i)T^{2} \) |
| 59 | \( 1 + (0.364 + 0.791i)T + (-38.3 + 44.8i)T^{2} \) |
| 61 | \( 1 + (3.04 - 0.624i)T + (56.0 - 23.9i)T^{2} \) |
| 67 | \( 1 + (-4.98 + 0.738i)T + (64.1 - 19.4i)T^{2} \) |
| 71 | \( 1 + (-0.178 - 0.0366i)T + (65.2 + 27.9i)T^{2} \) |
| 73 | \( 1 + (-1.04 + 1.89i)T + (-38.7 - 61.8i)T^{2} \) |
| 79 | \( 1 + (-0.463 + 0.296i)T + (33.0 - 71.7i)T^{2} \) |
| 83 | \( 1 + (-1.34 + 9.68i)T + (-79.8 - 22.5i)T^{2} \) |
| 89 | \( 1 + (-1.75 - 3.17i)T + (-47.3 + 75.3i)T^{2} \) |
| 97 | \( 1 + (-17.8 - 2.64i)T + (92.8 + 28.1i)T^{2} \) |
| 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
−10.16136614949692816122983385060, −9.418621564468054158623342031074, −8.310387943203843348884027514754, −7.53569103319870670331775732283, −6.57008576818262724283477707954, −5.63939616180581711630082943295, −5.32224599388547807545835071594, −3.24165683472333568605044152381, −1.99096039939591013202742997724, −0.44243831344332165105393672164,
1.87622245229451182095475184919, 2.63913944630092537216308174835, 4.50747920367983616731583401985, 5.18521340998199394719769945561, 6.22281216556173275240981591670, 7.41088115363514964591752345679, 8.030018331126255656426189774703, 9.503319322037169541959874666118, 9.943899811967729854447869768305, 10.21874618473377328209321492385
