L(s) = 1 | + (−2.20 − 1.27i)2-s + (2.24 + 3.88i)4-s + (0.817 + 2.08i)5-s + (−2.54 + 1.46i)7-s − 6.31i·8-s + (0.846 − 5.62i)10-s + (−0.317 + 0.550i)11-s + (3.60 − 0.0716i)13-s + 7.48·14-s + (−3.55 + 6.16i)16-s + (1.05 − 0.611i)17-s + (0.682 + 1.18i)19-s + (−6.24 + 7.83i)20-s + (1.40 − 0.808i)22-s + (−1.86 − 1.07i)23-s + ⋯ |
L(s) = 1 | + (−1.55 − 0.900i)2-s + (1.12 + 1.94i)4-s + (0.365 + 0.930i)5-s + (−0.961 + 0.555i)7-s − 2.23i·8-s + (0.267 − 1.78i)10-s + (−0.0957 + 0.165i)11-s + (0.999 − 0.0198i)13-s + 1.99·14-s + (−0.889 + 1.54i)16-s + (0.257 − 0.148i)17-s + (0.156 + 0.271i)19-s + (−1.39 + 1.75i)20-s + (0.298 − 0.172i)22-s + (−0.388 − 0.224i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 585 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.125 - 0.992i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 585 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.125 - 0.992i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.264087 + 0.299638i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.264087 + 0.299638i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 5 | \( 1 + (-0.817 - 2.08i)T \) |
| 13 | \( 1 + (-3.60 + 0.0716i)T \) |
good | 2 | \( 1 + (2.20 + 1.27i)T + (1 + 1.73i)T^{2} \) |
| 7 | \( 1 + (2.54 - 1.46i)T + (3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (0.317 - 0.550i)T + (-5.5 - 9.52i)T^{2} \) |
| 17 | \( 1 + (-1.05 + 0.611i)T + (8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-0.682 - 1.18i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (1.86 + 1.07i)T + (11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (1.5 - 2.59i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + 8.96T + 31T^{2} \) |
| 37 | \( 1 + (1.05 + 0.611i)T + (18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (4.98 - 8.62i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (-1.18 + 0.683i)T + (21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 - 6.16iT - 47T^{2} \) |
| 53 | \( 1 - 0.642iT - 53T^{2} \) |
| 59 | \( 1 + (3.79 + 6.57i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-1.13 - 1.96i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (6.95 + 4.01i)T + (33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (-1.31 - 2.28i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 - 10.3iT - 73T^{2} \) |
| 79 | \( 1 + 1.03T + 79T^{2} \) |
| 83 | \( 1 - 11.8iT - 83T^{2} \) |
| 89 | \( 1 + (-6.27 + 10.8i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (12.8 - 7.39i)T + (48.5 - 84.0i)T^{2} \) |
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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
−10.82888550802467557010647423224, −9.935006492626439300998563809626, −9.450923285590722863616013138784, −8.597247747602440707372029921967, −7.60309364391246117513242117664, −6.71397016311215844528179911411, −5.80545449519849992498412204152, −3.57727177497259332258716230240, −2.87450166157146223744966424214, −1.68584599691798898852558225611,
0.37296955454060633829143896227, 1.69862365798905887183173390712, 3.74667919993048902514850376913, 5.42864805045044561234459432514, 6.13235489346077769279542612888, 7.04405103937517373668392600554, 7.930616458202066438740015548982, 8.835959938811384564727554002886, 9.319499868198401740836535340975, 10.17235181902691206364484839740