L(s) = 1 | + (0.5 + 0.866i)2-s + (−0.499 + 0.866i)4-s + (−0.794 − 1.37i)5-s + (1.40 − 2.24i)7-s − 0.999·8-s + (0.794 − 1.37i)10-s + (0.794 − 1.37i)11-s − 4.81·13-s + (2.64 + 0.0963i)14-s + (−0.5 − 0.866i)16-s + (−2.69 + 4.67i)17-s + (−3.54 − 6.14i)19-s + 1.58·20-s + 1.58·22-s + (−0.150 − 0.260i)23-s + ⋯ |
L(s) = 1 | + (0.353 + 0.612i)2-s + (−0.249 + 0.433i)4-s + (−0.355 − 0.615i)5-s + (0.531 − 0.847i)7-s − 0.353·8-s + (0.251 − 0.434i)10-s + (0.239 − 0.414i)11-s − 1.33·13-s + (0.706 + 0.0257i)14-s + (−0.125 − 0.216i)16-s + (−0.654 + 1.13i)17-s + (−0.814 − 1.41i)19-s + 0.355·20-s + 0.338·22-s + (−0.0313 − 0.0542i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1134 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0996 + 0.995i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1134 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0996 + 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.9571699882\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9571699882\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.5 - 0.866i)T \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (-1.40 + 2.24i)T \) |
good | 5 | \( 1 + (0.794 + 1.37i)T + (-2.5 + 4.33i)T^{2} \) |
| 11 | \( 1 + (-0.794 + 1.37i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + 4.81T + 13T^{2} \) |
| 17 | \( 1 + (2.69 - 4.67i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (3.54 + 6.14i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (0.150 + 0.260i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + 8.27T + 29T^{2} \) |
| 31 | \( 1 + (-1.35 + 2.34i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (-0.5 - 0.866i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + 5.87T + 41T^{2} \) |
| 43 | \( 1 - 1.66T + 43T^{2} \) |
| 47 | \( 1 + (1.33 + 2.30i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (-2.44 + 4.23i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (3.23 - 5.60i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-2.23 - 3.87i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-5.02 + 8.70i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 - 12.7T + 71T^{2} \) |
| 73 | \( 1 + (-8.02 + 13.9i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (4.19 + 7.26i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + 2.36T + 83T^{2} \) |
| 89 | \( 1 + (-1.60 - 2.78i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + 1.42T + 97T^{2} \) |
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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
−9.383182665533292263337540379516, −8.591571170538832276275867015657, −7.918756039392911359242498570301, −7.09342651361578960602277079298, −6.35183166633152964357663755909, −5.08944172263743685607848463060, −4.52159226329993061107549219855, −3.72093124740181347776476123985, −2.15375555088186542980703910146, −0.34951405689446561372440713971,
1.88715868979233250334510859128, 2.67209960011429352830630890602, 3.82517997703292050499498052373, 4.83783695518978570808572298696, 5.55703976800695274111517555236, 6.73551465815983117512114274982, 7.50225287768629732968826552213, 8.479432527918774528431181485604, 9.444454625980450251061520200518, 10.00483079172165342574242178178