L(s) = 1 | + (0.5 − 0.866i)2-s + (−0.5 − 0.866i)3-s + (−0.499 − 0.866i)4-s + (−1.5 + 2.59i)5-s − 0.999·6-s + (0.5 − 2.59i)7-s − 0.999·8-s + (−0.499 + 0.866i)9-s + (1.5 + 2.59i)10-s + (2.5 + 4.33i)11-s + (−0.499 + 0.866i)12-s + (−2 − 1.73i)14-s + 3·15-s + (−0.5 + 0.866i)16-s + (2 + 3.46i)17-s + (0.499 + 0.866i)18-s + ⋯ |
L(s) = 1 | + (0.353 − 0.612i)2-s + (−0.288 − 0.499i)3-s + (−0.249 − 0.433i)4-s + (−0.670 + 1.16i)5-s − 0.408·6-s + (0.188 − 0.981i)7-s − 0.353·8-s + (−0.166 + 0.288i)9-s + (0.474 + 0.821i)10-s + (0.753 + 1.30i)11-s + (−0.144 + 0.249i)12-s + (−0.534 − 0.462i)14-s + 0.774·15-s + (−0.125 + 0.216i)16-s + (0.485 + 0.840i)17-s + (0.117 + 0.204i)18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.991 + 0.126i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.991 + 0.126i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.45931 - 0.0926083i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.45931 - 0.0926083i\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + (0.5 + 0.866i)T \) |
| 7 | \( 1 + (-0.5 + 2.59i)T \) |
| 23 | \( 1 + (0.5 - 0.866i)T \) |
good | 5 | \( 1 + (1.5 - 2.59i)T + (-2.5 - 4.33i)T^{2} \) |
| 11 | \( 1 + (-2.5 - 4.33i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + 13T^{2} \) |
| 17 | \( 1 + (-2 - 3.46i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-9.5 - 16.4i)T^{2} \) |
| 29 | \( 1 + 3T + 29T^{2} \) |
| 31 | \( 1 + (-4.5 - 7.79i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (-6 + 10.3i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 - 12T + 41T^{2} \) |
| 43 | \( 1 - 6T + 43T^{2} \) |
| 47 | \( 1 + (1 - 1.73i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (2.5 + 4.33i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (-4.5 - 7.79i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-1 + 1.73i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-1 - 1.73i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 - 6T + 71T^{2} \) |
| 73 | \( 1 + (-7 - 12.1i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (5.5 - 9.52i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + 17T + 83T^{2} \) |
| 89 | \( 1 + (5 - 8.66i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + 13T + 97T^{2} \) |
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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
−10.25261162515689573400733939698, −9.495037505427235443974712315924, −8.088741524543364875753841812108, −7.24599006001878626005770668970, −6.82450806400615382653206040504, −5.72125152001093631461818901784, −4.30963973791060101563727783530, −3.81663452723781749276112096808, −2.53069887300620699468188616683, −1.24029064146650868114045456993,
0.75326052828357931026428979232, 2.91819436233278651467542769673, 4.08377005163099850398828949352, 4.79171287602688970636966379969, 5.70981443157699516037835041513, 6.25770126867669343429437759305, 7.74471412427014847447529700448, 8.357840045885784651778141879406, 9.087920251240340788560753482955, 9.650407768669745372666202718997