| L(s) = 1 | + (1.04 + 0.948i)2-s + (−0.737 + 0.371i)3-s + (0.201 + 1.98i)4-s + (−0.947 − 2.13i)5-s + (−1.12 − 0.309i)6-s + (−1.69 − 1.01i)7-s + (−1.67 + 2.27i)8-s + (−1.38 + 1.86i)9-s + (1.03 − 3.14i)10-s + (−2.72 − 2.12i)11-s + (−0.886 − 1.39i)12-s + (−4.76 + 4.53i)13-s + (−0.814 − 2.67i)14-s + (1.49 + 1.22i)15-s + (−3.91 + 0.801i)16-s + (−3.71 − 4.52i)17-s + ⋯ |
| L(s) = 1 | + (0.741 + 0.670i)2-s + (−0.425 + 0.214i)3-s + (0.100 + 0.994i)4-s + (−0.423 − 0.956i)5-s + (−0.459 − 0.126i)6-s + (−0.640 − 0.383i)7-s + (−0.592 + 0.805i)8-s + (−0.460 + 0.620i)9-s + (0.326 − 0.993i)10-s + (−0.822 − 0.641i)11-s + (−0.255 − 0.401i)12-s + (−1.32 + 1.25i)13-s + (−0.217 − 0.714i)14-s + (0.385 + 0.316i)15-s + (−0.979 + 0.200i)16-s + (−0.900 − 1.09i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 512 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.904 + 0.427i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 512 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.904 + 0.427i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.0836984 - 0.373011i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0836984 - 0.373011i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-1.04 - 0.948i)T \) |
| good | 3 | \( 1 + (0.737 - 0.371i)T + (1.78 - 2.40i)T^{2} \) |
| 5 | \( 1 + (0.947 + 2.13i)T + (-3.35 + 3.70i)T^{2} \) |
| 7 | \( 1 + (1.69 + 1.01i)T + (3.29 + 6.17i)T^{2} \) |
| 11 | \( 1 + (2.72 + 2.12i)T + (2.67 + 10.6i)T^{2} \) |
| 13 | \( 1 + (4.76 - 4.53i)T + (0.637 - 12.9i)T^{2} \) |
| 17 | \( 1 + (3.71 + 4.52i)T + (-3.31 + 16.6i)T^{2} \) |
| 19 | \( 1 + (-1.33 - 7.71i)T + (-17.8 + 6.40i)T^{2} \) |
| 23 | \( 1 + (-6.89 - 3.26i)T + (14.5 + 17.7i)T^{2} \) |
| 29 | \( 1 + (0.747 + 0.423i)T + (14.9 + 24.8i)T^{2} \) |
| 31 | \( 1 + (-0.194 + 0.0386i)T + (28.6 - 11.8i)T^{2} \) |
| 37 | \( 1 + (0.729 + 0.461i)T + (15.8 + 33.4i)T^{2} \) |
| 41 | \( 1 + (5.86 + 6.47i)T + (-4.01 + 40.8i)T^{2} \) |
| 43 | \( 1 + (-0.934 - 2.82i)T + (-34.5 + 25.6i)T^{2} \) |
| 47 | \( 1 + (-6.68 - 3.57i)T + (26.1 + 39.0i)T^{2} \) |
| 53 | \( 1 + (-9.42 + 5.34i)T + (27.2 - 45.4i)T^{2} \) |
| 59 | \( 1 + (6.90 - 7.25i)T + (-2.89 - 58.9i)T^{2} \) |
| 61 | \( 1 + (-0.301 - 4.08i)T + (-60.3 + 8.95i)T^{2} \) |
| 67 | \( 1 + (3.84 + 3.31i)T + (9.83 + 66.2i)T^{2} \) |
| 71 | \( 1 + (-7.65 - 1.13i)T + (67.9 + 20.6i)T^{2} \) |
| 73 | \( 1 + (6.78 - 4.06i)T + (34.4 - 64.3i)T^{2} \) |
| 79 | \( 1 + (0.415 + 0.126i)T + (65.6 + 43.8i)T^{2} \) |
| 83 | \( 1 + (4.89 - 3.10i)T + (35.4 - 75.0i)T^{2} \) |
| 89 | \( 1 + (10.4 - 4.94i)T + (56.4 - 68.7i)T^{2} \) |
| 97 | \( 1 + (-5.74 - 3.83i)T + (37.1 + 89.6i)T^{2} \) |
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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
−11.73847191741018223495739282048, −10.69803357309719080082530293388, −9.446769452907059283841962375521, −8.590611170101020016982374859547, −7.61252425072310706673897740695, −6.85403312092072863832938640284, −5.51775029944675650075095269489, −4.98000046730074681753064250896, −4.03683832229582727405368094003, −2.64941844220154408134988809153,
0.17122198384059925398978042420, 2.67751092518500154155314876697, 3.05856124941495738091080070823, 4.68518158981243406526983726127, 5.56293378890042155907881631049, 6.70059680628009739574698476955, 7.18645139452431148006656316460, 8.828118736998766515933389001206, 9.838162368617537560678979991866, 10.73425977725761703978326654576
