| L(s) = 1 | + (0.959 − 0.281i)2-s + (−2.19 − 2.52i)3-s + (0.841 − 0.540i)4-s + (−0.453 − 3.15i)5-s + (−2.81 − 1.80i)6-s + (0.415 − 0.909i)7-s + (0.654 − 0.755i)8-s + (−1.16 + 8.11i)9-s + (−1.32 − 2.89i)10-s + (0.119 + 0.0350i)11-s + (−3.21 − 0.942i)12-s + (2.39 + 5.24i)13-s + (0.142 − 0.989i)14-s + (−6.98 + 8.06i)15-s + (0.415 − 0.909i)16-s + (−5.55 − 3.56i)17-s + ⋯ |
| L(s) = 1 | + (0.678 − 0.199i)2-s + (−1.26 − 1.45i)3-s + (0.420 − 0.270i)4-s + (−0.202 − 1.41i)5-s + (−1.14 − 0.738i)6-s + (0.157 − 0.343i)7-s + (0.231 − 0.267i)8-s + (−0.388 + 2.70i)9-s + (−0.418 − 0.917i)10-s + (0.0360 + 0.0105i)11-s + (−0.926 − 0.272i)12-s + (0.664 + 1.45i)13-s + (0.0380 − 0.264i)14-s + (−1.80 + 2.08i)15-s + (0.103 − 0.227i)16-s + (−1.34 − 0.865i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 322 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.990 + 0.136i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 322 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.990 + 0.136i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.0782717 - 1.13846i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0782717 - 1.13846i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.959 + 0.281i)T \) |
| 7 | \( 1 + (-0.415 + 0.909i)T \) |
| 23 | \( 1 + (0.765 + 4.73i)T \) |
| good | 3 | \( 1 + (2.19 + 2.52i)T + (-0.426 + 2.96i)T^{2} \) |
| 5 | \( 1 + (0.453 + 3.15i)T + (-4.79 + 1.40i)T^{2} \) |
| 11 | \( 1 + (-0.119 - 0.0350i)T + (9.25 + 5.94i)T^{2} \) |
| 13 | \( 1 + (-2.39 - 5.24i)T + (-8.51 + 9.82i)T^{2} \) |
| 17 | \( 1 + (5.55 + 3.56i)T + (7.06 + 15.4i)T^{2} \) |
| 19 | \( 1 + (-3.37 + 2.16i)T + (7.89 - 17.2i)T^{2} \) |
| 29 | \( 1 + (-2.36 - 1.51i)T + (12.0 + 26.3i)T^{2} \) |
| 31 | \( 1 + (-0.755 + 0.871i)T + (-4.41 - 30.6i)T^{2} \) |
| 37 | \( 1 + (-0.509 + 3.54i)T + (-35.5 - 10.4i)T^{2} \) |
| 41 | \( 1 + (-0.641 - 4.45i)T + (-39.3 + 11.5i)T^{2} \) |
| 43 | \( 1 + (0.301 + 0.348i)T + (-6.11 + 42.5i)T^{2} \) |
| 47 | \( 1 + 8.52T + 47T^{2} \) |
| 53 | \( 1 + (-0.0254 + 0.0557i)T + (-34.7 - 40.0i)T^{2} \) |
| 59 | \( 1 + (0.927 + 2.03i)T + (-38.6 + 44.5i)T^{2} \) |
| 61 | \( 1 + (3.07 - 3.55i)T + (-8.68 - 60.3i)T^{2} \) |
| 67 | \( 1 + (-10.2 + 3.01i)T + (56.3 - 36.2i)T^{2} \) |
| 71 | \( 1 + (-8.38 + 2.46i)T + (59.7 - 38.3i)T^{2} \) |
| 73 | \( 1 + (-6.91 + 4.44i)T + (30.3 - 66.4i)T^{2} \) |
| 79 | \( 1 + (0.159 + 0.349i)T + (-51.7 + 59.7i)T^{2} \) |
| 83 | \( 1 + (-2.14 + 14.8i)T + (-79.6 - 23.3i)T^{2} \) |
| 89 | \( 1 + (-4.05 - 4.67i)T + (-12.6 + 88.0i)T^{2} \) |
| 97 | \( 1 + (-0.965 - 6.71i)T + (-93.0 + 27.3i)T^{2} \) |
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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
−11.54506204270524589426900501751, −10.90105226967203288230244611342, −9.196670860718464975667930416896, −8.107191343739772581300128822715, −6.93455766817887487430673078394, −6.33698312501431691534079624215, −5.01584384082382066657475332392, −4.53120849884162452939731560434, −1.99960309417814322870952245713, −0.791781758272061090864451605728,
3.16282525359163273761577480961, 3.91696904713718271773627506853, 5.21645101254341004879331031994, 6.00094239615969809732152998682, 6.73536106827284049981016434521, 8.228245879243424023002968145726, 9.730934419664598082866529249238, 10.56286977877164153245999322283, 11.09007587214986939788121676952, 11.69817944583313757807910198100
