| 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 | \( 1 + (-5.11 + 5.90i)T + (-13.8 - 96.0i)T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.646172712157510240627560604757, −9.188253518540631956511949153483, −8.124897163975689272951781222059, −7.23687499435139930153246464196, −5.91778605355824891167650339517, −5.24270705641034586646846274614, −4.11314635011420371336491878041, −3.25151543613681005977977756299, −2.07794613866473092881473722167, −0.21541357731193098879918276031,
1.57296044931089617505384666394, 3.11074476069172212189294844309, 4.42357423671735162619923067466, 5.34376892889546690949460097657, 6.26029767737245252467018745004, 7.06747809972184674958438736680, 7.63927595107799418602223949510, 8.589905197144005943369175493343, 9.541788416380778248494791771062, 10.15012829249778828712678195003
