L(s) = 1 | + (1.32 + 0.959i)2-s + (0.206 + 0.634i)4-s + (2.42 − 1.76i)5-s + (−0.469 − 1.44i)7-s + (0.672 − 2.07i)8-s + 4.89·10-s + (−3.29 + 0.334i)11-s + (3.32 + 2.41i)13-s + (0.766 − 2.35i)14-s + (3.95 − 2.87i)16-s + (−2.16 + 1.57i)17-s + (−2.64 + 8.14i)19-s + (1.61 + 1.17i)20-s + (−4.68 − 2.72i)22-s − 4.20·23-s + ⋯ |
L(s) = 1 | + (0.934 + 0.678i)2-s + (0.103 + 0.317i)4-s + (1.08 − 0.787i)5-s + (−0.177 − 0.546i)7-s + (0.237 − 0.731i)8-s + 1.54·10-s + (−0.994 + 0.100i)11-s + (0.921 + 0.669i)13-s + (0.204 − 0.630i)14-s + (0.988 − 0.718i)16-s + (−0.526 + 0.382i)17-s + (−0.607 + 1.86i)19-s + (0.361 + 0.262i)20-s + (−0.997 − 0.580i)22-s − 0.876·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 297 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.992 - 0.124i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 297 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.992 - 0.124i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.23981 + 0.140048i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.23981 + 0.140048i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 11 | \( 1 + (3.29 - 0.334i)T \) |
good | 2 | \( 1 + (-1.32 - 0.959i)T + (0.618 + 1.90i)T^{2} \) |
| 5 | \( 1 + (-2.42 + 1.76i)T + (1.54 - 4.75i)T^{2} \) |
| 7 | \( 1 + (0.469 + 1.44i)T + (-5.66 + 4.11i)T^{2} \) |
| 13 | \( 1 + (-3.32 - 2.41i)T + (4.01 + 12.3i)T^{2} \) |
| 17 | \( 1 + (2.16 - 1.57i)T + (5.25 - 16.1i)T^{2} \) |
| 19 | \( 1 + (2.64 - 8.14i)T + (-15.3 - 11.1i)T^{2} \) |
| 23 | \( 1 + 4.20T + 23T^{2} \) |
| 29 | \( 1 + (-0.598 - 1.84i)T + (-23.4 + 17.0i)T^{2} \) |
| 31 | \( 1 + (-5.57 - 4.04i)T + (9.57 + 29.4i)T^{2} \) |
| 37 | \( 1 + (0.208 + 0.643i)T + (-29.9 + 21.7i)T^{2} \) |
| 41 | \( 1 + (-2.04 + 6.29i)T + (-33.1 - 24.0i)T^{2} \) |
| 43 | \( 1 - 0.0153T + 43T^{2} \) |
| 47 | \( 1 + (1.19 - 3.67i)T + (-38.0 - 27.6i)T^{2} \) |
| 53 | \( 1 + (-0.0180 - 0.0130i)T + (16.3 + 50.4i)T^{2} \) |
| 59 | \( 1 + (4.17 + 12.8i)T + (-47.7 + 34.6i)T^{2} \) |
| 61 | \( 1 + (9.51 - 6.91i)T + (18.8 - 58.0i)T^{2} \) |
| 67 | \( 1 + 7.15T + 67T^{2} \) |
| 71 | \( 1 + (-2.70 + 1.96i)T + (21.9 - 67.5i)T^{2} \) |
| 73 | \( 1 + (2.56 + 7.89i)T + (-59.0 + 42.9i)T^{2} \) |
| 79 | \( 1 + (-4.38 - 3.18i)T + (24.4 + 75.1i)T^{2} \) |
| 83 | \( 1 + (-8.82 + 6.41i)T + (25.6 - 78.9i)T^{2} \) |
| 89 | \( 1 + 9.84T + 89T^{2} \) |
| 97 | \( 1 + (-12.7 - 9.24i)T + (29.9 + 92.2i)T^{2} \) |
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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
−12.30614519568995408927316019254, −10.56816686429589153468685993850, −10.05677866080968567485690030662, −8.877540950362969317149574221784, −7.78336638391757835347981525277, −6.38713330963881208945365964814, −5.87611329601892314867652164041, −4.80121456621377528634296263365, −3.80562148536171194203882265522, −1.66991077088864182985231493459,
2.39462593565039792200881627935, 2.89644463615225899183094791556, 4.52868613352065748197862368250, 5.65942834243171544139895308981, 6.41424046615979285411586682339, 7.936416025604759241386344394043, 9.029737716009843322781499934486, 10.26010446360730997073212893213, 10.89752281125885269731444937561, 11.72667980381758899877991978651