L(s) = 1 | − 2-s + (0.5 − 0.866i)3-s + 4-s + (−0.5 − 0.866i)5-s + (−0.5 + 0.866i)6-s + (−0.695 + 1.20i)7-s − 8-s + (−0.499 − 0.866i)9-s + (0.5 + 0.866i)10-s + (−2.61 − 4.53i)11-s + (0.5 − 0.866i)12-s + (−0.390 − 0.676i)13-s + (0.695 − 1.20i)14-s − 0.999·15-s + 16-s + (−1.80 + 3.12i)17-s + ⋯ |
L(s) = 1 | − 0.707·2-s + (0.288 − 0.499i)3-s + 0.5·4-s + (−0.223 − 0.387i)5-s + (−0.204 + 0.353i)6-s + (−0.262 + 0.455i)7-s − 0.353·8-s + (−0.166 − 0.288i)9-s + (0.158 + 0.273i)10-s + (−0.789 − 1.36i)11-s + (0.144 − 0.249i)12-s + (−0.108 − 0.187i)13-s + (0.185 − 0.321i)14-s − 0.258·15-s + 0.250·16-s + (−0.437 + 0.758i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.879 - 0.475i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.879 - 0.475i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.0198073 + 0.0782176i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.0198073 + 0.0782176i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 3 | \( 1 + (-0.5 + 0.866i)T \) |
| 5 | \( 1 + (0.5 + 0.866i)T \) |
| 31 | \( 1 + (5.54 + 0.527i)T \) |
good | 7 | \( 1 + (0.695 - 1.20i)T + (-3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (2.61 + 4.53i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (0.390 + 0.676i)T + (-6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 + (1.80 - 3.12i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (3.72 - 6.45i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 - 2.39T + 23T^{2} \) |
| 29 | \( 1 + 2.84T + 29T^{2} \) |
| 37 | \( 1 + (-2.80 + 4.85i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (5.50 + 9.54i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (3.92 - 6.79i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 - 1.17T + 47T^{2} \) |
| 53 | \( 1 + (-3.81 - 6.60i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (2.08 - 3.61i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + 9.45T + 61T^{2} \) |
| 67 | \( 1 + (2.80 + 4.85i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (-1.60 - 2.78i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (-1 - 1.73i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (3.92 - 6.79i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (6.93 + 12.0i)T + (-41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + 9.21T + 89T^{2} \) |
| 97 | \( 1 - 13.2T + 97T^{2} \) |
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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
−9.322448285814415160321790000663, −8.620622626344795642909902572157, −8.122579258695617440014675050229, −7.29693125398827872155148124977, −6.06422796288955784546991584416, −5.61649871989578060818825006330, −3.92009140314902500324853090463, −2.86528406495466777377553116107, −1.66261993017766728665428778203, −0.04299369751476599777834216751,
2.11664973311103435555120863408, 3.04990005647335337851843356199, 4.36305101590847907561901960653, 5.15107489570603082739643358572, 6.79356362402632121127436257253, 7.10469743170991911950960880693, 8.080935309689468668418033847239, 9.054781870513933615600127524660, 9.702610957220854957632268747197, 10.42192585536930964225105127895