| 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 | \( 1 + (1.21 - 0.882i)T + (29.9 - 92.2i)T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−12.73325313603969579184911857548, −11.87988356423732047425947957808, −10.97611231495990210945161462850, −10.28911120820932503095894500756, −9.299231046524535742511740319763, −7.81371276936030481610151480357, −6.77117178982815128573226386239, −5.09864949159950742781207146426, −3.91396595650554236474837334782, −2.08959845916282485844296962097,
1.22786740220948346480124704589, 4.22323358205086127576466658121, 5.45710638119489757845118112524, 6.33458672652040128067310709995, 7.921397933557687455571201214919, 8.522335688744759306742220518208, 9.615563685170127233446338075854, 11.35701731242394527239408762338, 11.62794365915299264393793094882, 13.31006606409504743272694517554
