L(s) = 1 | + (−0.423 − 0.333i)2-s + (0.814 + 0.580i)3-s + (−0.402 − 1.66i)4-s + (2.64 − 0.510i)5-s + (−0.151 − 0.517i)6-s + (2.13 − 1.55i)7-s + (−0.830 + 1.81i)8-s + (0.327 + 0.945i)9-s + (−1.29 − 0.666i)10-s + (3.14 + 3.99i)11-s + (0.635 − 1.58i)12-s + (−0.763 − 1.18i)13-s + (−1.42 − 0.0535i)14-s + (2.45 + 1.11i)15-s + (−2.08 + 1.07i)16-s + (−2.02 − 1.92i)17-s + ⋯ |
L(s) = 1 | + (−0.299 − 0.235i)2-s + (0.470 + 0.334i)3-s + (−0.201 − 0.830i)4-s + (1.18 − 0.228i)5-s + (−0.0620 − 0.211i)6-s + (0.808 − 0.588i)7-s + (−0.293 + 0.643i)8-s + (0.109 + 0.315i)9-s + (−0.408 − 0.210i)10-s + (0.947 + 1.20i)11-s + (0.183 − 0.458i)12-s + (−0.211 − 0.329i)13-s + (−0.380 − 0.0143i)14-s + (0.633 + 0.289i)15-s + (−0.520 + 0.268i)16-s + (−0.490 − 0.467i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.752 + 0.658i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.752 + 0.658i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.64822 - 0.619676i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.64822 - 0.619676i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-0.814 - 0.580i)T \) |
| 7 | \( 1 + (-2.13 + 1.55i)T \) |
| 23 | \( 1 + (0.824 + 4.72i)T \) |
good | 2 | \( 1 + (0.423 + 0.333i)T + (0.471 + 1.94i)T^{2} \) |
| 5 | \( 1 + (-2.64 + 0.510i)T + (4.64 - 1.85i)T^{2} \) |
| 11 | \( 1 + (-3.14 - 3.99i)T + (-2.59 + 10.6i)T^{2} \) |
| 13 | \( 1 + (0.763 + 1.18i)T + (-5.40 + 11.8i)T^{2} \) |
| 17 | \( 1 + (2.02 + 1.92i)T + (0.808 + 16.9i)T^{2} \) |
| 19 | \( 1 + (1.09 - 1.04i)T + (0.904 - 18.9i)T^{2} \) |
| 29 | \( 1 + (-2.58 + 0.760i)T + (24.3 - 15.6i)T^{2} \) |
| 31 | \( 1 + (-0.211 - 2.21i)T + (-30.4 + 5.86i)T^{2} \) |
| 37 | \( 1 + (4.81 - 1.66i)T + (29.0 - 22.8i)T^{2} \) |
| 41 | \( 1 + (-2.53 + 2.19i)T + (5.83 - 40.5i)T^{2} \) |
| 43 | \( 1 + (0.960 - 0.438i)T + (28.1 - 32.4i)T^{2} \) |
| 47 | \( 1 + (-4.74 - 2.73i)T + (23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (3.13 + 0.149i)T + (52.7 + 5.03i)T^{2} \) |
| 59 | \( 1 + (0.420 - 0.815i)T + (-34.2 - 48.0i)T^{2} \) |
| 61 | \( 1 + (-1.51 - 2.12i)T + (-19.9 + 57.6i)T^{2} \) |
| 67 | \( 1 + (5.20 + 12.9i)T + (-48.4 + 46.2i)T^{2} \) |
| 71 | \( 1 + (1.34 - 9.33i)T + (-68.1 - 20.0i)T^{2} \) |
| 73 | \( 1 + (12.1 - 2.94i)T + (64.8 - 33.4i)T^{2} \) |
| 79 | \( 1 + (13.4 - 0.640i)T + (78.6 - 7.50i)T^{2} \) |
| 83 | \( 1 + (-4.65 + 5.36i)T + (-11.8 - 82.1i)T^{2} \) |
| 89 | \( 1 + (13.8 + 1.32i)T + (87.3 + 16.8i)T^{2} \) |
| 97 | \( 1 + (-11.5 - 13.3i)T + (-13.8 + 96.0i)T^{2} \) |
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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
−10.42036909389950997787884017075, −10.08865738823534764348603390017, −9.227066298095448878156354794352, −8.576486895235036169507974709972, −7.22170012983961005899981486126, −6.16765526335170471972541199417, −4.99263490321323680630903955447, −4.35287919597999079751342759975, −2.31395197935616231528904382678, −1.42831738055000502900441000852,
1.72187327936588335909293176589, 2.92910385114621682468282033276, 4.19016925992672534627208829959, 5.71527096489858620220269636526, 6.51651398806143560182858358609, 7.55308446047914200589129650466, 8.705957008740942256655256385931, 8.879624361041916722323807585220, 9.910254281754216770460155996873, 11.22584269752385245891488787974