L(s) = 1 | + (0.5 + 0.866i)3-s + (0.5 − 0.866i)5-s + (−2 + 1.73i)7-s + (1 − 1.73i)9-s + (2.5 + 4.33i)11-s − 2·13-s + 0.999·15-s + (1.5 + 2.59i)17-s + (1.5 − 2.59i)19-s + (−2.5 − 0.866i)21-s + (0.5 − 0.866i)23-s + (2 + 3.46i)25-s + 5·27-s − 10·29-s + (5.5 + 9.52i)31-s + ⋯ |
L(s) = 1 | + (0.288 + 0.499i)3-s + (0.223 − 0.387i)5-s + (−0.755 + 0.654i)7-s + (0.333 − 0.577i)9-s + (0.753 + 1.30i)11-s − 0.554·13-s + 0.258·15-s + (0.363 + 0.630i)17-s + (0.344 − 0.596i)19-s + (−0.545 − 0.188i)21-s + (0.104 − 0.180i)23-s + (0.400 + 0.692i)25-s + 0.962·27-s − 1.85·29-s + (0.987 + 1.71i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1148 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.266 - 0.963i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1148 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.266 - 0.963i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.707899915\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.707899915\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 7 | \( 1 + (2 - 1.73i)T \) |
| 41 | \( 1 + T \) |
good | 3 | \( 1 + (-0.5 - 0.866i)T + (-1.5 + 2.59i)T^{2} \) |
| 5 | \( 1 + (-0.5 + 0.866i)T + (-2.5 - 4.33i)T^{2} \) |
| 11 | \( 1 + (-2.5 - 4.33i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + 2T + 13T^{2} \) |
| 17 | \( 1 + (-1.5 - 2.59i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-1.5 + 2.59i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-0.5 + 0.866i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + 10T + 29T^{2} \) |
| 31 | \( 1 + (-5.5 - 9.52i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (3.5 - 6.06i)T + (-18.5 - 32.0i)T^{2} \) |
| 43 | \( 1 - 8T + 43T^{2} \) |
| 47 | \( 1 + (-3.5 + 6.06i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (-5.5 - 9.52i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (-3.5 - 6.06i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-0.5 + 0.866i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (3.5 + 6.06i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + 12T + 71T^{2} \) |
| 73 | \( 1 + (-2.5 - 4.33i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (4.5 - 7.79i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + 83T^{2} \) |
| 89 | \( 1 + (-1.5 + 2.59i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 - 2T + 97T^{2} \) |
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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
−9.843194629798360725010746390678, −9.133155327122756312966308034976, −8.797733400188825772003259141151, −7.29178221018063956457510863140, −6.76578586891479422091967928991, −5.66156370713247587826042427706, −4.74424878987062399324459939309, −3.83094836733668712886190321263, −2.82829821525619053579961886191, −1.46761439352581439677629190486,
0.77109845396288235273662873895, 2.26879067946146416498864663867, 3.30585998392088565279308606388, 4.20603108264576271409834714921, 5.60436204347377688606277733808, 6.33134364212133292015079266860, 7.30529306012363243290356955598, 7.71631351194631518032281633100, 8.866689842337609554039035466799, 9.664303771330357933669447041705