L(s) = 1 | + (−0.142 + 0.989i)2-s + (−0.415 − 0.909i)3-s + (−0.959 − 0.281i)4-s + (0.0335 + 0.0386i)5-s + (0.959 − 0.281i)6-s + (−0.841 − 0.540i)7-s + (0.415 − 0.909i)8-s + (−0.654 + 0.755i)9-s + (−0.0430 + 0.0276i)10-s + (0.0735 + 0.511i)11-s + (0.142 + 0.989i)12-s + (−2.90 + 1.86i)13-s + (0.654 − 0.755i)14-s + (0.0212 − 0.0465i)15-s + (0.841 + 0.540i)16-s + (2.55 − 0.751i)17-s + ⋯ |
L(s) = 1 | + (−0.100 + 0.699i)2-s + (−0.239 − 0.525i)3-s + (−0.479 − 0.140i)4-s + (0.0149 + 0.0172i)5-s + (0.391 − 0.115i)6-s + (−0.317 − 0.204i)7-s + (0.146 − 0.321i)8-s + (−0.218 + 0.251i)9-s + (−0.0136 + 0.00874i)10-s + (0.0221 + 0.154i)11-s + (0.0410 + 0.285i)12-s + (−0.806 + 0.518i)13-s + (0.175 − 0.201i)14-s + (0.00548 − 0.0120i)15-s + (0.210 + 0.135i)16-s + (0.620 − 0.182i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.933 - 0.357i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.933 - 0.357i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.0805175 + 0.434962i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.0805175 + 0.434962i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.142 - 0.989i)T \) |
| 3 | \( 1 + (0.415 + 0.909i)T \) |
| 7 | \( 1 + (0.841 + 0.540i)T \) |
| 23 | \( 1 + (1.23 - 4.63i)T \) |
good | 5 | \( 1 + (-0.0335 - 0.0386i)T + (-0.711 + 4.94i)T^{2} \) |
| 11 | \( 1 + (-0.0735 - 0.511i)T + (-10.5 + 3.09i)T^{2} \) |
| 13 | \( 1 + (2.90 - 1.86i)T + (5.40 - 11.8i)T^{2} \) |
| 17 | \( 1 + (-2.55 + 0.751i)T + (14.3 - 9.19i)T^{2} \) |
| 19 | \( 1 + (1.27 + 0.375i)T + (15.9 + 10.2i)T^{2} \) |
| 29 | \( 1 + (6.48 - 1.90i)T + (24.3 - 15.6i)T^{2} \) |
| 31 | \( 1 + (2.66 - 5.82i)T + (-20.3 - 23.4i)T^{2} \) |
| 37 | \( 1 + (4.42 - 5.10i)T + (-5.26 - 36.6i)T^{2} \) |
| 41 | \( 1 + (-0.924 - 1.06i)T + (-5.83 + 40.5i)T^{2} \) |
| 43 | \( 1 + (-5.11 - 11.2i)T + (-28.1 + 32.4i)T^{2} \) |
| 47 | \( 1 + 6.09T + 47T^{2} \) |
| 53 | \( 1 + (9.23 + 5.93i)T + (22.0 + 48.2i)T^{2} \) |
| 59 | \( 1 + (-6.76 + 4.34i)T + (24.5 - 53.6i)T^{2} \) |
| 61 | \( 1 + (5.30 - 11.6i)T + (-39.9 - 46.1i)T^{2} \) |
| 67 | \( 1 + (1.24 - 8.64i)T + (-64.2 - 18.8i)T^{2} \) |
| 71 | \( 1 + (0.357 - 2.48i)T + (-68.1 - 20.0i)T^{2} \) |
| 73 | \( 1 + (9.27 + 2.72i)T + (61.4 + 39.4i)T^{2} \) |
| 79 | \( 1 + (4.57 - 2.94i)T + (32.8 - 71.8i)T^{2} \) |
| 83 | \( 1 + (-5.44 + 6.28i)T + (-11.8 - 82.1i)T^{2} \) |
| 89 | \( 1 + (6.34 + 13.8i)T + (-58.2 + 67.2i)T^{2} \) |
| 97 | \( 1 + (-5.11 - 5.90i)T + (-13.8 + 96.0i)T^{2} \) |
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
−10.15012829249778828712678195003, −9.541788416380778248494791771062, −8.589905197144005943369175493343, −7.63927595107799418602223949510, −7.06747809972184674958438736680, −6.26029767737245252467018745004, −5.34376892889546690949460097657, −4.42357423671735162619923067466, −3.11074476069172212189294844309, −1.57296044931089617505384666394,
0.21541357731193098879918276031, 2.07794613866473092881473722167, 3.25151543613681005977977756299, 4.11314635011420371336491878041, 5.24270705641034586646846274614, 5.91778605355824891167650339517, 7.23687499435139930153246464196, 8.124897163975689272951781222059, 9.188253518540631956511949153483, 9.646172712157510240627560604757