L(s) = 1 | + i·2-s + (0.500 + 1.65i)3-s − 4-s + (0.866 − 0.5i)5-s + (−1.65 + 0.500i)6-s + (−1.28 + 2.22i)7-s − i·8-s + (−2.49 + 1.65i)9-s + (0.5 + 0.866i)10-s + (2.56 + 4.44i)11-s + (−0.500 − 1.65i)12-s + (4.59 − 2.65i)13-s + (−2.22 − 1.28i)14-s + (1.26 + 1.18i)15-s + 16-s + (−2.59 + 4.49i)17-s + ⋯ |
L(s) = 1 | + 0.707i·2-s + (0.288 + 0.957i)3-s − 0.5·4-s + (0.387 − 0.223i)5-s + (−0.676 + 0.204i)6-s + (−0.486 + 0.842i)7-s − 0.353i·8-s + (−0.833 + 0.553i)9-s + (0.158 + 0.273i)10-s + (0.774 + 1.34i)11-s + (−0.144 − 0.478i)12-s + (1.27 − 0.735i)13-s + (−0.595 − 0.343i)14-s + (0.325 + 0.306i)15-s + 0.250·16-s + (−0.629 + 1.08i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.995 - 0.0942i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.995 - 0.0942i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.0719362 + 1.52362i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.0719362 + 1.52362i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - iT \) |
| 3 | \( 1 + (-0.500 - 1.65i)T \) |
| 5 | \( 1 + (-0.866 + 0.5i)T \) |
| 31 | \( 1 + (-0.790 + 5.51i)T \) |
good | 7 | \( 1 + (1.28 - 2.22i)T + (-3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (-2.56 - 4.44i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (-4.59 + 2.65i)T + (6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 + (2.59 - 4.49i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (0.264 - 0.458i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 - 3.03T + 23T^{2} \) |
| 29 | \( 1 + 8.44T + 29T^{2} \) |
| 37 | \( 1 + (-0.580 - 0.335i)T + (18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (9.85 - 5.68i)T + (20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (6.69 + 3.86i)T + (21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + 6.75iT - 47T^{2} \) |
| 53 | \( 1 + (-0.822 - 1.42i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (-10.4 - 6.03i)T + (29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 - 7.06iT - 61T^{2} \) |
| 67 | \( 1 + (-3.98 - 6.89i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (-8.45 + 4.88i)T + (35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + (3.21 - 1.85i)T + (36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-8.75 - 5.05i)T + (39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (3.97 + 6.87i)T + (-41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 - 14.8T + 89T^{2} \) |
| 97 | \( 1 + 6.46T + 97T^{2} \) |
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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
−10.12014758477660012174730431490, −9.547097565314970935464093795025, −8.764062292986862409016230336325, −8.302642411983471592019275443726, −6.95146128648598809528530994038, −6.01066235009127201750229493883, −5.39421194597903645741129340575, −4.28123848699858468082066457099, −3.48200763176424403676658253403, −1.97667264220241052066210975486,
0.71250293465955208733462787279, 1.80922107323469125649301231945, 3.22997268018113807645540817218, 3.74336639604709235890683045143, 5.34523419108091133310973267375, 6.58499214023689150881750153439, 6.79054967148667227497993923866, 8.164185387472139287037407169522, 8.979522308747031599083279681348, 9.457211698380346585580633169418