| L(s) = 1 | + (−1.41 − 0.00887i)2-s + (0.0605 + 2.46i)3-s + (1.99 + 0.0251i)4-s + (0.349 + 1.26i)5-s + (−0.0637 − 3.49i)6-s + (0.896 − 1.89i)7-s + (−2.82 − 0.0532i)8-s + (−3.09 + 0.151i)9-s + (−0.482 − 1.78i)10-s + (4.25 + 2.99i)11-s + (0.0592 + 4.93i)12-s + (4.83 − 0.595i)13-s + (−1.28 + 2.67i)14-s + (−3.09 + 0.938i)15-s + (3.99 + 0.100i)16-s + (0.174 − 0.574i)17-s + ⋯ |
| L(s) = 1 | + (−0.999 − 0.00627i)2-s + (0.0349 + 1.42i)3-s + (0.999 + 0.0125i)4-s + (0.156 + 0.564i)5-s + (−0.0260 − 1.42i)6-s + (0.338 − 0.716i)7-s + (−0.999 − 0.0188i)8-s + (−1.03 + 0.0506i)9-s + (−0.152 − 0.565i)10-s + (1.28 + 0.904i)11-s + (0.0170 + 1.42i)12-s + (1.33 − 0.165i)13-s + (−0.343 + 0.714i)14-s + (−0.798 + 0.242i)15-s + (0.999 + 0.0251i)16-s + (0.0422 − 0.139i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 512 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.247 - 0.968i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 512 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.247 - 0.968i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.673415 + 0.867245i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.673415 + 0.867245i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.41 + 0.00887i)T \) |
| good | 3 | \( 1 + (-0.0605 - 2.46i)T + (-2.99 + 0.147i)T^{2} \) |
| 5 | \( 1 + (-0.349 - 1.26i)T + (-4.28 + 2.57i)T^{2} \) |
| 7 | \( 1 + (-0.896 + 1.89i)T + (-4.44 - 5.41i)T^{2} \) |
| 11 | \( 1 + (-4.25 - 2.99i)T + (3.70 + 10.3i)T^{2} \) |
| 13 | \( 1 + (-4.83 + 0.595i)T + (12.6 - 3.15i)T^{2} \) |
| 17 | \( 1 + (-0.174 + 0.574i)T + (-14.1 - 9.44i)T^{2} \) |
| 19 | \( 1 + (0.452 - 6.13i)T + (-18.7 - 2.78i)T^{2} \) |
| 23 | \( 1 + (-1.31 + 0.974i)T + (6.67 - 22.0i)T^{2} \) |
| 29 | \( 1 + (6.47 - 1.45i)T + (26.2 - 12.3i)T^{2} \) |
| 31 | \( 1 + (4.94 + 7.40i)T + (-11.8 + 28.6i)T^{2} \) |
| 37 | \( 1 + (0.410 + 0.815i)T + (-22.0 + 29.7i)T^{2} \) |
| 41 | \( 1 + (8.55 + 5.12i)T + (19.3 + 36.1i)T^{2} \) |
| 43 | \( 1 + (1.32 + 1.39i)T + (-2.10 + 42.9i)T^{2} \) |
| 47 | \( 1 + (-5.16 - 4.23i)T + (9.16 + 46.0i)T^{2} \) |
| 53 | \( 1 + (4.29 + 0.964i)T + (47.9 + 22.6i)T^{2} \) |
| 59 | \( 1 + (1.31 - 10.6i)T + (-57.2 - 14.3i)T^{2} \) |
| 61 | \( 1 + (4.74 + 10.6i)T + (-40.9 + 45.1i)T^{2} \) |
| 67 | \( 1 + (-0.339 + 0.880i)T + (-49.6 - 44.9i)T^{2} \) |
| 71 | \( 1 + (-9.42 - 10.3i)T + (-6.95 + 70.6i)T^{2} \) |
| 73 | \( 1 + (1.30 + 2.76i)T + (-46.3 + 56.4i)T^{2} \) |
| 79 | \( 1 + (0.106 - 1.07i)T + (-77.4 - 15.4i)T^{2} \) |
| 83 | \( 1 + (-5.18 + 10.3i)T + (-49.4 - 66.6i)T^{2} \) |
| 89 | \( 1 + (0.269 + 0.200i)T + (25.8 + 85.1i)T^{2} \) |
| 97 | \( 1 + (12.4 + 2.47i)T + (89.6 + 37.1i)T^{2} \) |
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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
−10.79448518067818528066474411782, −10.28375791942176665805760738269, −9.464045143292383273442376538862, −8.827378594147546609916559545542, −7.67374481431397511033115353932, −6.71512071188298805768612750276, −5.70084372755413104478479727670, −4.11946591067257461663184137732, −3.50545348910371152208057362472, −1.64027026014462689915969139050,
1.04123419548083502711628078391, 1.81689656413365347744112100264, 3.34063215627857703899558470177, 5.44244454620686988376386907832, 6.41264594331829791536466643140, 6.95970029779492588111894187464, 8.174990625208651458664483632989, 8.836228766849642185225631478671, 9.183480196860354011013661028346, 10.93492376774403585852738894446
