| L(s) = 1 | + (0.891 + 0.453i)2-s + (0.832 − 1.51i)3-s + (0.587 + 0.809i)4-s + (0.0831 + 2.23i)5-s + (1.43 − 0.975i)6-s + (0.0556 + 0.0556i)7-s + (0.156 + 0.987i)8-s + (−1.61 − 2.52i)9-s + (−0.940 + 2.02i)10-s + (1.04 − 0.340i)11-s + (1.71 − 0.219i)12-s + (−2.31 − 4.54i)13-s + (0.0243 + 0.0748i)14-s + (3.46 + 1.73i)15-s + (−0.309 + 0.951i)16-s + (−3.10 + 0.491i)17-s + ⋯ |
| L(s) = 1 | + (0.630 + 0.321i)2-s + (0.480 − 0.876i)3-s + (0.293 + 0.404i)4-s + (0.0371 + 0.999i)5-s + (0.584 − 0.398i)6-s + (0.0210 + 0.0210i)7-s + (0.0553 + 0.349i)8-s + (−0.538 − 0.842i)9-s + (−0.297 + 0.641i)10-s + (0.315 − 0.102i)11-s + (0.495 − 0.0633i)12-s + (−0.641 − 1.25i)13-s + (0.00649 + 0.0200i)14-s + (0.894 + 0.447i)15-s + (−0.0772 + 0.237i)16-s + (−0.752 + 0.119i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.994 - 0.106i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 150 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.994 - 0.106i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.69335 + 0.0908255i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.69335 + 0.0908255i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.891 - 0.453i)T \) |
| 3 | \( 1 + (-0.832 + 1.51i)T \) |
| 5 | \( 1 + (-0.0831 - 2.23i)T \) |
| good | 7 | \( 1 + (-0.0556 - 0.0556i)T + 7iT^{2} \) |
| 11 | \( 1 + (-1.04 + 0.340i)T + (8.89 - 6.46i)T^{2} \) |
| 13 | \( 1 + (2.31 + 4.54i)T + (-7.64 + 10.5i)T^{2} \) |
| 17 | \( 1 + (3.10 - 0.491i)T + (16.1 - 5.25i)T^{2} \) |
| 19 | \( 1 + (0.824 - 1.13i)T + (-5.87 - 18.0i)T^{2} \) |
| 23 | \( 1 + (1.13 - 2.22i)T + (-13.5 - 18.6i)T^{2} \) |
| 29 | \( 1 + (5.66 - 4.11i)T + (8.96 - 27.5i)T^{2} \) |
| 31 | \( 1 + (-7.72 - 5.61i)T + (9.57 + 29.4i)T^{2} \) |
| 37 | \( 1 + (-9.23 + 4.70i)T + (21.7 - 29.9i)T^{2} \) |
| 41 | \( 1 + (3.55 + 1.15i)T + (33.1 + 24.0i)T^{2} \) |
| 43 | \( 1 + (1.00 - 1.00i)T - 43iT^{2} \) |
| 47 | \( 1 + (-1.98 + 12.5i)T + (-44.6 - 14.5i)T^{2} \) |
| 53 | \( 1 + (-6.70 - 1.06i)T + (50.4 + 16.3i)T^{2} \) |
| 59 | \( 1 + (2.03 - 6.26i)T + (-47.7 - 34.6i)T^{2} \) |
| 61 | \( 1 + (-1.23 - 3.79i)T + (-49.3 + 35.8i)T^{2} \) |
| 67 | \( 1 + (1.04 + 6.57i)T + (-63.7 + 20.7i)T^{2} \) |
| 71 | \( 1 + (-7.51 - 10.3i)T + (-21.9 + 67.5i)T^{2} \) |
| 73 | \( 1 + (-1.18 - 0.602i)T + (42.9 + 59.0i)T^{2} \) |
| 79 | \( 1 + (-5.85 - 8.05i)T + (-24.4 + 75.1i)T^{2} \) |
| 83 | \( 1 + (-0.137 - 0.871i)T + (-78.9 + 25.6i)T^{2} \) |
| 89 | \( 1 + (1.01 + 3.11i)T + (-72.0 + 52.3i)T^{2} \) |
| 97 | \( 1 + (10.2 + 1.62i)T + (92.2 + 29.9i)T^{2} \) |
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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
−13.20328608463496683356960246705, −12.26008947297571055639758465779, −11.28389453758430094898888815873, −10.06885634805877900321759223957, −8.545524574350767459765001682382, −7.49474303076785166628193576106, −6.70463222952190996833338587290, −5.60008528686862769196051837068, −3.62067172398940819967976935430, −2.43572826272047822825563611439,
2.30733667732368400043440266476, 4.21794209146715277155830872016, 4.69861410788173160892649365200, 6.22653985562820487650067217027, 7.945951717525049267346545838027, 9.215572576832621841714685277745, 9.735187836836194751693809395122, 11.18176562038012583111328980071, 11.94050161661622622186885312475, 13.17197567625892592837159497674
