| L(s) = 1 | + (−0.959 − 0.281i)2-s + (0.755 + 0.654i)3-s + (0.841 + 0.540i)4-s + (0.480 − 3.34i)5-s + (−0.540 − 0.841i)6-s + (−2.08 − 1.63i)7-s + (−0.654 − 0.755i)8-s + (0.142 + 0.989i)9-s + (−1.40 + 3.07i)10-s + (−0.0493 − 0.168i)11-s + (0.281 + 0.959i)12-s + (−2.65 − 1.21i)13-s + (1.53 + 2.15i)14-s + (2.55 − 2.21i)15-s + (0.415 + 0.909i)16-s + (0.603 − 0.387i)17-s + ⋯ |
| L(s) = 1 | + (−0.678 − 0.199i)2-s + (0.436 + 0.378i)3-s + (0.420 + 0.270i)4-s + (0.214 − 1.49i)5-s + (−0.220 − 0.343i)6-s + (−0.787 − 0.616i)7-s + (−0.231 − 0.267i)8-s + (0.0474 + 0.329i)9-s + (−0.443 + 0.970i)10-s + (−0.0148 − 0.0506i)11-s + (0.0813 + 0.276i)12-s + (−0.735 − 0.335i)13-s + (0.411 + 0.575i)14-s + (0.658 − 0.570i)15-s + (0.103 + 0.227i)16-s + (0.146 − 0.0940i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.970 + 0.240i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.970 + 0.240i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.0715229 - 0.586796i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0715229 - 0.586796i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.959 + 0.281i)T \) |
| 3 | \( 1 + (-0.755 - 0.654i)T \) |
| 7 | \( 1 + (2.08 + 1.63i)T \) |
| 23 | \( 1 + (3.94 - 2.72i)T \) |
| good | 5 | \( 1 + (-0.480 + 3.34i)T + (-4.79 - 1.40i)T^{2} \) |
| 11 | \( 1 + (0.0493 + 0.168i)T + (-9.25 + 5.94i)T^{2} \) |
| 13 | \( 1 + (2.65 + 1.21i)T + (8.51 + 9.82i)T^{2} \) |
| 17 | \( 1 + (-0.603 + 0.387i)T + (7.06 - 15.4i)T^{2} \) |
| 19 | \( 1 + (3.02 + 1.94i)T + (7.89 + 17.2i)T^{2} \) |
| 29 | \( 1 + (-1.49 + 0.962i)T + (12.0 - 26.3i)T^{2} \) |
| 31 | \( 1 + (3.42 - 2.97i)T + (4.41 - 30.6i)T^{2} \) |
| 37 | \( 1 + (7.95 - 1.14i)T + (35.5 - 10.4i)T^{2} \) |
| 41 | \( 1 + (-7.90 - 1.13i)T + (39.3 + 11.5i)T^{2} \) |
| 43 | \( 1 + (-0.300 - 0.260i)T + (6.11 + 42.5i)T^{2} \) |
| 47 | \( 1 - 1.90iT - 47T^{2} \) |
| 53 | \( 1 + (4.39 - 2.00i)T + (34.7 - 40.0i)T^{2} \) |
| 59 | \( 1 + (-0.693 - 0.316i)T + (38.6 + 44.5i)T^{2} \) |
| 61 | \( 1 + (8.23 + 9.50i)T + (-8.68 + 60.3i)T^{2} \) |
| 67 | \( 1 + (-2.74 + 9.34i)T + (-56.3 - 36.2i)T^{2} \) |
| 71 | \( 1 + (0.959 + 0.281i)T + (59.7 + 38.3i)T^{2} \) |
| 73 | \( 1 + (-4.71 + 7.34i)T + (-30.3 - 66.4i)T^{2} \) |
| 79 | \( 1 + (0.704 + 0.321i)T + (51.7 + 59.7i)T^{2} \) |
| 83 | \( 1 + (0.697 + 4.85i)T + (-79.6 + 23.3i)T^{2} \) |
| 89 | \( 1 + (-6.55 + 7.56i)T + (-12.6 - 88.0i)T^{2} \) |
| 97 | \( 1 + (-1.81 + 12.6i)T + (-93.0 - 27.3i)T^{2} \) |
| 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
−9.478439479870948189807277030429, −9.044651611189742349506262778835, −8.134386734164693514648897824679, −7.42190157409852740065105386501, −6.25374681502983587721493912401, −5.11919518184921058875226079647, −4.23154160065731400269524519691, −3.13235185488423734036024734664, −1.75559127946485292045395696825, −0.29979731057629468592811235117,
2.10906366659076167441746412643, 2.72274320981019920436114914057, 3.85062556139816870653905439382, 5.65732905734710400512483363933, 6.45965589201365229676127716401, 6.99230365997397076000570054828, 7.81676393967063465636439289437, 8.776973401873197648137309679157, 9.614020986189018494835863933475, 10.23380400558169514349329169432
