| L(s) = 1 | + (−0.309 + 0.951i)2-s + (−0.809 − 0.587i)3-s + (−0.809 − 0.587i)4-s + (0.00655 − 2.23i)5-s + (0.809 − 0.587i)6-s + 2.63·7-s + (0.809 − 0.587i)8-s + (0.309 + 0.951i)9-s + (2.12 + 0.697i)10-s + (1.93 − 5.96i)11-s + (0.309 + 0.951i)12-s + (0.697 + 2.14i)13-s + (−0.815 + 2.51i)14-s + (−1.31 + 1.80i)15-s + (0.309 + 0.951i)16-s + (−1.31 + 0.955i)17-s + ⋯ |
| L(s) = 1 | + (−0.218 + 0.672i)2-s + (−0.467 − 0.339i)3-s + (−0.404 − 0.293i)4-s + (0.00293 − 0.999i)5-s + (0.330 − 0.239i)6-s + 0.997·7-s + (0.286 − 0.207i)8-s + (0.103 + 0.317i)9-s + (0.671 + 0.220i)10-s + (0.584 − 1.79i)11-s + (0.0892 + 0.274i)12-s + (0.193 + 0.595i)13-s + (−0.217 + 0.670i)14-s + (−0.340 + 0.466i)15-s + (0.0772 + 0.237i)16-s + (−0.319 + 0.231i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.927 + 0.373i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 150 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.927 + 0.373i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.899413 - 0.174307i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.899413 - 0.174307i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.309 - 0.951i)T \) |
| 3 | \( 1 + (0.809 + 0.587i)T \) |
| 5 | \( 1 + (-0.00655 + 2.23i)T \) |
| good | 7 | \( 1 - 2.63T + 7T^{2} \) |
| 11 | \( 1 + (-1.93 + 5.96i)T + (-8.89 - 6.46i)T^{2} \) |
| 13 | \( 1 + (-0.697 - 2.14i)T + (-10.5 + 7.64i)T^{2} \) |
| 17 | \( 1 + (1.31 - 0.955i)T + (5.25 - 16.1i)T^{2} \) |
| 19 | \( 1 + (-1 + 0.726i)T + (5.87 - 18.0i)T^{2} \) |
| 23 | \( 1 + (1.38 - 4.25i)T + (-18.6 - 13.5i)T^{2} \) |
| 29 | \( 1 + (5.55 + 4.03i)T + (8.96 + 27.5i)T^{2} \) |
| 31 | \( 1 + (-4.75 + 3.45i)T + (9.57 - 29.4i)T^{2} \) |
| 37 | \( 1 + (-2.49 - 7.68i)T + (-29.9 + 21.7i)T^{2} \) |
| 41 | \( 1 + (-3.55 - 10.9i)T + (-33.1 + 24.0i)T^{2} \) |
| 43 | \( 1 + 1.97T + 43T^{2} \) |
| 47 | \( 1 + (5.25 + 3.81i)T + (14.5 + 44.6i)T^{2} \) |
| 53 | \( 1 + (-5.66 - 4.11i)T + (16.3 + 50.4i)T^{2} \) |
| 59 | \( 1 + (-2.79 - 8.59i)T + (-47.7 + 34.6i)T^{2} \) |
| 61 | \( 1 + (0.933 - 2.87i)T + (-49.3 - 35.8i)T^{2} \) |
| 67 | \( 1 + (-8.12 + 5.90i)T + (20.7 - 63.7i)T^{2} \) |
| 71 | \( 1 + (-7.90 - 5.74i)T + (21.9 + 67.5i)T^{2} \) |
| 73 | \( 1 + (3.70 - 11.4i)T + (-59.0 - 42.9i)T^{2} \) |
| 79 | \( 1 + (3.03 + 2.20i)T + (24.4 + 75.1i)T^{2} \) |
| 83 | \( 1 + (-6.77 + 4.92i)T + (25.6 - 78.9i)T^{2} \) |
| 89 | \( 1 + (0.120 - 0.370i)T + (-72.0 - 52.3i)T^{2} \) |
| 97 | \( 1 + (1.21 + 0.882i)T + (29.9 + 92.2i)T^{2} \) |
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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
−13.31006606409504743272694517554, −11.62794365915299264393793094882, −11.35701731242394527239408762338, −9.615563685170127233446338075854, −8.522335688744759306742220518208, −7.921397933557687455571201214919, −6.33458672652040128067310709995, −5.45710638119489757845118112524, −4.22323358205086127576466658121, −1.22786740220948346480124704589,
2.08959845916282485844296962097, 3.91396595650554236474837334782, 5.09864949159950742781207146426, 6.77117178982815128573226386239, 7.81371276936030481610151480357, 9.299231046524535742511740319763, 10.28911120820932503095894500756, 10.97611231495990210945161462850, 11.87988356423732047425947957808, 12.73325313603969579184911857548
