L(s) = 1 | + 0.202i·2-s − 2.47i·3-s + 1.95·4-s + (−1.53 + 1.62i)5-s + 0.501·6-s − 5.06i·7-s + 0.801i·8-s − 3.14·9-s + (−0.329 − 0.310i)10-s + 11-s − 4.85i·12-s − 1.72i·13-s + 1.02·14-s + (4.03 + 3.80i)15-s + 3.75·16-s − 2.65i·17-s + ⋯ |
L(s) = 1 | + 0.143i·2-s − 1.43i·3-s + 0.979·4-s + (−0.686 + 0.727i)5-s + 0.204·6-s − 1.91i·7-s + 0.283i·8-s − 1.04·9-s + (−0.104 − 0.0981i)10-s + 0.301·11-s − 1.40i·12-s − 0.479i·13-s + 0.273·14-s + (1.04 + 0.981i)15-s + 0.938·16-s − 0.642i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.686 + 0.727i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.686 + 0.727i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.604757262\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.604757262\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 + (1.53 - 1.62i)T \) |
| 11 | \( 1 - T \) |
| 19 | \( 1 + T \) |
good | 2 | \( 1 - 0.202iT - 2T^{2} \) |
| 3 | \( 1 + 2.47iT - 3T^{2} \) |
| 7 | \( 1 + 5.06iT - 7T^{2} \) |
| 13 | \( 1 + 1.72iT - 13T^{2} \) |
| 17 | \( 1 + 2.65iT - 17T^{2} \) |
| 23 | \( 1 - 6.75iT - 23T^{2} \) |
| 29 | \( 1 + 10.1T + 29T^{2} \) |
| 31 | \( 1 - 6.13T + 31T^{2} \) |
| 37 | \( 1 + 7.00iT - 37T^{2} \) |
| 41 | \( 1 + 3.72T + 41T^{2} \) |
| 43 | \( 1 - 0.935iT - 43T^{2} \) |
| 47 | \( 1 + 3.29iT - 47T^{2} \) |
| 53 | \( 1 + 0.668iT - 53T^{2} \) |
| 59 | \( 1 - 3.35T + 59T^{2} \) |
| 61 | \( 1 - 13.2T + 61T^{2} \) |
| 67 | \( 1 - 12.8iT - 67T^{2} \) |
| 71 | \( 1 - 0.833T + 71T^{2} \) |
| 73 | \( 1 + 1.38iT - 73T^{2} \) |
| 79 | \( 1 + 11.3T + 79T^{2} \) |
| 83 | \( 1 + 14.3iT - 83T^{2} \) |
| 89 | \( 1 - 1.51T + 89T^{2} \) |
| 97 | \( 1 - 14.9iT - 97T^{2} \) |
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
−9.879218070460919163457063908932, −8.207073786269685713711406979818, −7.49333798009110600546014800719, −7.21197346354575067509900183693, −6.72111613330234854754805566848, −5.68700171006204707959109443822, −4.01087832065677063325517054533, −3.16994759224339543300214230800, −1.87642269890350815160960608969, −0.70106495450076700557380054277,
1.93649284272224245237408260148, 3.06682071482610233421442281895, 4.04845107906878561378842977443, 4.98282003835786038685711029197, 5.78403965634971507713796492869, 6.66532188061995988587591016773, 8.128396514915478558742237002868, 8.684470805051194297069843405404, 9.395669816131151207828671635627, 10.16196797446133005692512156118