L(s) = 1 | + (−0.5 + 0.866i)2-s + (−0.0752 − 0.130i)3-s + (−0.499 − 0.866i)4-s + (0.125 − 2.23i)5-s + 0.150·6-s + (−2.53 − 0.742i)7-s + 0.999·8-s + (1.48 − 2.57i)9-s + (1.87 + 1.22i)10-s + (0.899 + 3.19i)11-s + (−0.0752 + 0.130i)12-s − 2.51i·13-s + (1.91 − 1.82i)14-s + (−0.300 + 0.151i)15-s + (−0.5 + 0.866i)16-s + (−1.99 + 1.14i)17-s + ⋯ |
L(s) = 1 | + (−0.353 + 0.612i)2-s + (−0.0434 − 0.0752i)3-s + (−0.249 − 0.433i)4-s + (0.0560 − 0.998i)5-s + 0.0614·6-s + (−0.959 − 0.280i)7-s + 0.353·8-s + (0.496 − 0.859i)9-s + (0.591 + 0.387i)10-s + (0.271 + 0.962i)11-s + (−0.0217 + 0.0376i)12-s − 0.697i·13-s + (0.511 − 0.488i)14-s + (−0.0775 + 0.0391i)15-s + (−0.125 + 0.216i)16-s + (−0.482 + 0.278i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 770 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.656 + 0.754i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 770 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.656 + 0.754i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.204060 - 0.448153i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.204060 - 0.448153i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.5 - 0.866i)T \) |
| 5 | \( 1 + (-0.125 + 2.23i)T \) |
| 7 | \( 1 + (2.53 + 0.742i)T \) |
| 11 | \( 1 + (-0.899 - 3.19i)T \) |
good | 3 | \( 1 + (0.0752 + 0.130i)T + (-1.5 + 2.59i)T^{2} \) |
| 13 | \( 1 + 2.51iT - 13T^{2} \) |
| 17 | \( 1 + (1.99 - 1.14i)T + (8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (3.16 - 5.47i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (4.94 + 2.85i)T + (11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + 6.85iT - 29T^{2} \) |
| 31 | \( 1 + (-1.82 + 1.05i)T + (15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (9.79 + 5.65i)T + (18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 - 0.710T + 41T^{2} \) |
| 43 | \( 1 + 2.99T + 43T^{2} \) |
| 47 | \( 1 + (3.34 - 5.78i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (3.78 - 2.18i)T + (26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (-7.75 + 4.47i)T + (29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (5.17 - 8.96i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-6.84 + 3.94i)T + (33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + 11.7T + 71T^{2} \) |
| 73 | \( 1 + (-11.5 + 6.65i)T + (36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (7.29 + 4.21i)T + (39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + 8.40iT - 83T^{2} \) |
| 89 | \( 1 + (-3.02 - 1.74i)T + (44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 - 9.73T + 97T^{2} \) |
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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
−9.932150731220213012155542730149, −9.171657201426608498548824053862, −8.302187839741585565911721517779, −7.44557407450755439863097086621, −6.40521877218353801249919286556, −5.89049988274812043838613434015, −4.48817933632950138181613025220, −3.80689758878331769585317038447, −1.81215112622178280299233696086, −0.26993836326328218010719445068,
1.97934566282434066585356825155, 3.00689402126005217405926583168, 3.90875113996489279701501069764, 5.21864582628560906855125824913, 6.58821615204932430405331552912, 6.95714559932410802188644542854, 8.268414342878840303575146568947, 9.071888541656507242959068279037, 9.936076378588273356557032945205, 10.60516061710722417356673823178