| L(s) = 1 | + (0.959 + 0.281i)2-s + (−0.654 + 0.755i)3-s + (0.841 + 0.540i)4-s + (−0.220 + 1.53i)5-s + (−0.841 + 0.540i)6-s + (−0.415 − 0.909i)7-s + (0.654 + 0.755i)8-s + (−0.142 − 0.989i)9-s + (−0.642 + 1.40i)10-s + (−5.05 + 1.48i)11-s + (−0.959 + 0.281i)12-s + (−0.667 + 1.46i)13-s + (−0.142 − 0.989i)14-s + (−1.01 − 1.16i)15-s + (0.415 + 0.909i)16-s + (−4.54 + 2.92i)17-s + ⋯ |
| L(s) = 1 | + (0.678 + 0.199i)2-s + (−0.378 + 0.436i)3-s + (0.420 + 0.270i)4-s + (−0.0984 + 0.684i)5-s + (−0.343 + 0.220i)6-s + (−0.157 − 0.343i)7-s + (0.231 + 0.267i)8-s + (−0.0474 − 0.329i)9-s + (−0.203 + 0.444i)10-s + (−1.52 + 0.447i)11-s + (−0.276 + 0.0813i)12-s + (−0.185 + 0.405i)13-s + (−0.0380 − 0.264i)14-s + (−0.261 − 0.301i)15-s + (0.103 + 0.227i)16-s + (−1.10 + 0.709i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.995 - 0.0960i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.995 - 0.0960i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.0467567 + 0.970908i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0467567 + 0.970908i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.959 - 0.281i)T \) |
| 3 | \( 1 + (0.654 - 0.755i)T \) |
| 7 | \( 1 + (0.415 + 0.909i)T \) |
| 23 | \( 1 + (3.22 + 3.54i)T \) |
| good | 5 | \( 1 + (0.220 - 1.53i)T + (-4.79 - 1.40i)T^{2} \) |
| 11 | \( 1 + (5.05 - 1.48i)T + (9.25 - 5.94i)T^{2} \) |
| 13 | \( 1 + (0.667 - 1.46i)T + (-8.51 - 9.82i)T^{2} \) |
| 17 | \( 1 + (4.54 - 2.92i)T + (7.06 - 15.4i)T^{2} \) |
| 19 | \( 1 + (1.57 + 1.01i)T + (7.89 + 17.2i)T^{2} \) |
| 29 | \( 1 + (-1.09 + 0.706i)T + (12.0 - 26.3i)T^{2} \) |
| 31 | \( 1 + (-4.95 - 5.72i)T + (-4.41 + 30.6i)T^{2} \) |
| 37 | \( 1 + (0.455 + 3.17i)T + (-35.5 + 10.4i)T^{2} \) |
| 41 | \( 1 + (0.616 - 4.28i)T + (-39.3 - 11.5i)T^{2} \) |
| 43 | \( 1 + (5.69 - 6.57i)T + (-6.11 - 42.5i)T^{2} \) |
| 47 | \( 1 + 5.56T + 47T^{2} \) |
| 53 | \( 1 + (0.814 + 1.78i)T + (-34.7 + 40.0i)T^{2} \) |
| 59 | \( 1 + (-0.315 + 0.690i)T + (-38.6 - 44.5i)T^{2} \) |
| 61 | \( 1 + (-0.651 - 0.752i)T + (-8.68 + 60.3i)T^{2} \) |
| 67 | \( 1 + (6.69 + 1.96i)T + (56.3 + 36.2i)T^{2} \) |
| 71 | \( 1 + (3.53 + 1.03i)T + (59.7 + 38.3i)T^{2} \) |
| 73 | \( 1 + (-9.88 - 6.35i)T + (30.3 + 66.4i)T^{2} \) |
| 79 | \( 1 + (-3.06 + 6.71i)T + (-51.7 - 59.7i)T^{2} \) |
| 83 | \( 1 + (0.588 + 4.09i)T + (-79.6 + 23.3i)T^{2} \) |
| 89 | \( 1 + (-0.112 + 0.130i)T + (-12.6 - 88.0i)T^{2} \) |
| 97 | \( 1 + (1.25 - 8.71i)T + (-93.0 - 27.3i)T^{2} \) |
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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
−10.55571815510162637266846969873, −9.944717418865633662583290557131, −8.625232696861764622199911796815, −7.76113946361928105756948589292, −6.73545209870223891616749136764, −6.28914575613075795014162065479, −4.96538582196462030765049153361, −4.46034173291668042341640214265, −3.24531660171737387304005148970, −2.26218635413848353213400513255,
0.34301056509464564773517872483, 2.09047081110731255457760320146, 3.05758797135311259229149405248, 4.51085064641215277077657606370, 5.22812769827598358760844367689, 5.93368329801964837187856321317, 6.93800674687236367404211775912, 7.947001518864890169368697646695, 8.603833476727571970923367048211, 9.804442604832946146073128886128
