| L(s) = 1 | + (0.841 − 0.540i)2-s + (−0.673 + 0.197i)3-s + (0.415 − 0.909i)4-s + (0.654 + 0.755i)5-s + (−0.459 + 0.530i)6-s + (−2.18 + 1.40i)7-s + (−0.142 − 0.989i)8-s + (−2.10 + 1.35i)9-s + (0.959 + 0.281i)10-s + (0.515 + 0.595i)11-s + (−0.0998 + 0.694i)12-s + (−0.545 + 3.79i)13-s + (−1.08 + 2.36i)14-s + (−0.590 − 0.379i)15-s + (−0.654 − 0.755i)16-s + (1.85 + 4.05i)17-s + ⋯ |
| L(s) = 1 | + (0.594 − 0.382i)2-s + (−0.388 + 0.114i)3-s + (0.207 − 0.454i)4-s + (0.292 + 0.337i)5-s + (−0.187 + 0.216i)6-s + (−0.826 + 0.531i)7-s + (−0.0503 − 0.349i)8-s + (−0.703 + 0.451i)9-s + (0.303 + 0.0890i)10-s + (0.155 + 0.179i)11-s + (−0.0288 + 0.200i)12-s + (−0.151 + 1.05i)13-s + (−0.288 + 0.632i)14-s + (−0.152 − 0.0979i)15-s + (−0.163 − 0.188i)16-s + (0.449 + 0.983i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 670 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.333 - 0.942i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 670 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.333 - 0.942i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.14432 + 0.808792i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.14432 + 0.808792i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.841 + 0.540i)T \) |
| 5 | \( 1 + (-0.654 - 0.755i)T \) |
| 67 | \( 1 + (-6.47 - 5.00i)T \) |
| good | 3 | \( 1 + (0.673 - 0.197i)T + (2.52 - 1.62i)T^{2} \) |
| 7 | \( 1 + (2.18 - 1.40i)T + (2.90 - 6.36i)T^{2} \) |
| 11 | \( 1 + (-0.515 - 0.595i)T + (-1.56 + 10.8i)T^{2} \) |
| 13 | \( 1 + (0.545 - 3.79i)T + (-12.4 - 3.66i)T^{2} \) |
| 17 | \( 1 + (-1.85 - 4.05i)T + (-11.1 + 12.8i)T^{2} \) |
| 19 | \( 1 + (-5.31 - 3.41i)T + (7.89 + 17.2i)T^{2} \) |
| 23 | \( 1 + (0.758 - 0.222i)T + (19.3 - 12.4i)T^{2} \) |
| 29 | \( 1 + 4.08T + 29T^{2} \) |
| 31 | \( 1 + (0.771 + 5.36i)T + (-29.7 + 8.73i)T^{2} \) |
| 37 | \( 1 + 8.64T + 37T^{2} \) |
| 41 | \( 1 + (-0.674 - 1.47i)T + (-26.8 + 30.9i)T^{2} \) |
| 43 | \( 1 + (-4.09 - 8.95i)T + (-28.1 + 32.4i)T^{2} \) |
| 47 | \( 1 + (-4.76 + 1.39i)T + (39.5 - 25.4i)T^{2} \) |
| 53 | \( 1 + (0.442 - 0.968i)T + (-34.7 - 40.0i)T^{2} \) |
| 59 | \( 1 + (-0.178 - 1.24i)T + (-56.6 + 16.6i)T^{2} \) |
| 61 | \( 1 + (-7.22 + 8.34i)T + (-8.68 - 60.3i)T^{2} \) |
| 71 | \( 1 + (-4.57 + 10.0i)T + (-46.4 - 53.6i)T^{2} \) |
| 73 | \( 1 + (2.57 - 2.97i)T + (-10.3 - 72.2i)T^{2} \) |
| 79 | \( 1 + (0.888 - 6.18i)T + (-75.7 - 22.2i)T^{2} \) |
| 83 | \( 1 + (2.30 + 2.66i)T + (-11.8 + 82.1i)T^{2} \) |
| 89 | \( 1 + (6.15 + 1.80i)T + (74.8 + 48.1i)T^{2} \) |
| 97 | \( 1 + 13.2T + 97T^{2} \) |
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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
−10.82261858803389008070930478766, −9.850192201449333015544925093929, −9.318049244519473540086952127046, −8.040983074227203501395966930472, −6.84706391643697730906682902300, −5.93532482977553343163661174430, −5.43981156365869990870078436410, −4.06912223168349351693951019762, −3.05001999166635983153732939728, −1.87447587936192811695540937994,
0.63407107128261886056046098674, 2.88262459948124846490770206739, 3.66064895791673240230660782910, 5.26547306056932469757545159632, 5.56566552793485794469470143827, 6.79404511845244715519474129506, 7.35695401832782464793349333640, 8.622481067512128861555311686551, 9.445597402248268124663837226120, 10.36789920961399544172223273751
