L(s) = 1 | + (−0.866 − 0.5i)2-s + (−1.60 + 0.923i)3-s + (0.499 + 0.866i)4-s + (−2.14 − 1.23i)5-s + 1.84·6-s − 0.999i·8-s + (0.207 − 0.358i)9-s + (1.23 + 2.14i)10-s + (3.29 + 0.397i)11-s + (−1.60 − 0.923i)12-s − 1.96·13-s + 4.56·15-s + (−0.5 + 0.866i)16-s + (2.82 + 4.89i)17-s + (−0.358 + 0.207i)18-s + (0.453 − 0.785i)19-s + ⋯ |
L(s) = 1 | + (−0.612 − 0.353i)2-s + (−0.923 + 0.533i)3-s + (0.249 + 0.433i)4-s + (−0.957 − 0.552i)5-s + 0.754·6-s − 0.353i·8-s + (0.0690 − 0.119i)9-s + (0.390 + 0.676i)10-s + (0.992 + 0.119i)11-s + (−0.461 − 0.266i)12-s − 0.544·13-s + 1.17·15-s + (−0.125 + 0.216i)16-s + (0.685 + 1.18i)17-s + (−0.0845 + 0.0488i)18-s + (0.104 − 0.180i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1078 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.858 + 0.512i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1078 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.858 + 0.512i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.1227560543\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.1227560543\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.866 + 0.5i)T \) |
| 7 | \( 1 \) |
| 11 | \( 1 + (-3.29 - 0.397i)T \) |
good | 3 | \( 1 + (1.60 - 0.923i)T + (1.5 - 2.59i)T^{2} \) |
| 5 | \( 1 + (2.14 + 1.23i)T + (2.5 + 4.33i)T^{2} \) |
| 13 | \( 1 + 1.96T + 13T^{2} \) |
| 17 | \( 1 + (-2.82 - 4.89i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-0.453 + 0.785i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-0.468 + 0.811i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 - 7.50iT - 29T^{2} \) |
| 31 | \( 1 + (4.08 - 2.35i)T + (15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (-4.81 + 8.33i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + 6.93T + 41T^{2} \) |
| 43 | \( 1 - 6.80iT - 43T^{2} \) |
| 47 | \( 1 + (3.46 + 2.00i)T + (23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (3.28 + 5.68i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (-0.836 + 0.482i)T + (29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-5.95 + 10.3i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (3.36 + 5.81i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + 11.5T + 71T^{2} \) |
| 73 | \( 1 + (5.43 + 9.40i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (7.34 + 4.24i)T + (39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + 4.83T + 83T^{2} \) |
| 89 | \( 1 + (11.0 + 6.36i)T + (44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + 3.82iT - 97T^{2} \) |
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
−9.639493186964323759555852640969, −8.763803647360091901640430047931, −8.046133094481544761459986977699, −7.14891553356542858586900403442, −6.13346238596027023116998224500, −5.07162720109341393329624394743, −4.24950380935533121149558315882, −3.37568844915329441164206766464, −1.57693191950967335542647425368, −0.089668020910666141341854045943,
1.17902052593376712882524860714, 2.92597712331050228529883782698, 4.13195504878554086708319930864, 5.38097428580987401698925054513, 6.19077433337978275332396427063, 7.10336409517423164015479250792, 7.40639449949455819063590073890, 8.436020467891507739950487570823, 9.457061990609359102847499541441, 10.14279427669452190901742256900