L(s) = 1 | + (−1.5 − 0.866i)2-s − 2·3-s + (0.5 + 0.866i)4-s + (−1.5 + 2.59i)5-s + (3 + 1.73i)6-s + (−3 − 1.73i)7-s + 1.73i·8-s + 9-s + (4.5 − 2.59i)10-s − 3.46i·11-s + (−1 − 1.73i)12-s + (1 − 1.73i)13-s + (3 + 5.19i)14-s + (3 − 5.19i)15-s + (2.49 − 4.33i)16-s + (−6 + 3.46i)17-s + ⋯ |
L(s) = 1 | + (−1.06 − 0.612i)2-s − 1.15·3-s + (0.250 + 0.433i)4-s + (−0.670 + 1.16i)5-s + (1.22 + 0.707i)6-s + (−1.13 − 0.654i)7-s + 0.612i·8-s + 0.333·9-s + (1.42 − 0.821i)10-s − 1.04i·11-s + (−0.288 − 0.500i)12-s + (0.277 − 0.480i)13-s + (0.801 + 1.38i)14-s + (0.774 − 1.34i)15-s + (0.624 − 1.08i)16-s + (−1.45 + 0.840i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 61 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.855 - 0.518i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 61 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.855 - 0.518i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 61 | \( 1 + (0.5 - 7.79i)T \) |
good | 2 | \( 1 + (1.5 + 0.866i)T + (1 + 1.73i)T^{2} \) |
| 3 | \( 1 + 2T + 3T^{2} \) |
| 5 | \( 1 + (1.5 - 2.59i)T + (-2.5 - 4.33i)T^{2} \) |
| 7 | \( 1 + (3 + 1.73i)T + (3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + 3.46iT - 11T^{2} \) |
| 13 | \( 1 + (-1 + 1.73i)T + (-6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 + (6 - 3.46i)T + (8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-1 - 1.73i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 - 23T^{2} \) |
| 29 | \( 1 + (1.5 - 0.866i)T + (14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (9 - 5.19i)T + (15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + 1.73iT - 37T^{2} \) |
| 41 | \( 1 - 3T + 41T^{2} \) |
| 43 | \( 1 + (-3 - 1.73i)T + (21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (6 + 10.3i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + 5.19iT - 53T^{2} \) |
| 59 | \( 1 + (3 + 1.73i)T + (29.5 + 51.0i)T^{2} \) |
| 67 | \( 1 + (3 + 1.73i)T + (33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (9 - 5.19i)T + (35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + (-3.5 - 6.06i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (9 + 5.19i)T + (39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (3 - 5.19i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + 15.5iT - 89T^{2} \) |
| 97 | \( 1 + (0.5 + 0.866i)T + (-48.5 + 84.0i)T^{2} \) |
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
−14.49703356369637803913610510678, −13.00367585485339188716691372910, −11.49025016979964945891749175245, −10.86378928719409194637113667134, −10.30865493469148430556997888510, −8.687154848939297440421022196735, −7.08316854214303469120456888516, −5.92144739334591805205480698882, −3.39144853322135670810655197720, 0,
4.55343076693636250565205258201, 6.17446311783826414281828765443, 7.32351152279063211893662906148, 8.934124914792456645089787817877, 9.486771794996276417784994054962, 11.22175428149875369070283455099, 12.35773274924212323012558365673, 13.00669108642035085648456205411, 15.45398636081071759679123171310