| L(s) = 1 | + (−0.499 − 1.32i)2-s + (−0.688 + 2.08i)3-s + (−1.50 + 1.32i)4-s + (−0.920 + 2.38i)5-s + (3.09 − 0.130i)6-s + (0.865 − 1.44i)7-s + (2.49 + 1.32i)8-s + (−1.45 − 1.07i)9-s + (3.61 + 0.0252i)10-s + (0.412 + 3.34i)11-s + (−1.72 − 4.03i)12-s + (−0.0702 + 2.86i)13-s + (−2.34 − 0.423i)14-s + (−4.33 − 3.55i)15-s + (0.503 − 3.96i)16-s + (−4.44 − 5.41i)17-s + ⋯ |
| L(s) = 1 | + (−0.353 − 0.935i)2-s + (−0.397 + 1.20i)3-s + (−0.750 + 0.661i)4-s + (−0.411 + 1.06i)5-s + (1.26 − 0.0533i)6-s + (0.327 − 0.545i)7-s + (0.883 + 0.468i)8-s + (−0.485 − 0.359i)9-s + (1.14 + 0.00798i)10-s + (0.124 + 1.00i)11-s + (−0.497 − 1.16i)12-s + (−0.0194 + 0.793i)13-s + (−0.626 − 0.113i)14-s + (−1.11 − 0.918i)15-s + (0.125 − 0.992i)16-s + (−1.07 − 1.31i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 512 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.772 - 0.635i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 512 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.772 - 0.635i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.190326 + 0.530674i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.190326 + 0.530674i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.499 + 1.32i)T \) |
| good | 3 | \( 1 + (0.688 - 2.08i)T + (-2.40 - 1.78i)T^{2} \) |
| 5 | \( 1 + (0.920 - 2.38i)T + (-3.70 - 3.35i)T^{2} \) |
| 7 | \( 1 + (-0.865 + 1.44i)T + (-3.29 - 6.17i)T^{2} \) |
| 11 | \( 1 + (-0.412 - 3.34i)T + (-10.6 + 2.67i)T^{2} \) |
| 13 | \( 1 + (0.0702 - 2.86i)T + (-12.9 - 0.637i)T^{2} \) |
| 17 | \( 1 + (4.44 + 5.41i)T + (-3.31 + 16.6i)T^{2} \) |
| 19 | \( 1 + (-1.05 - 0.741i)T + (6.40 + 17.8i)T^{2} \) |
| 23 | \( 1 + (0.0809 - 0.171i)T + (-14.5 - 17.7i)T^{2} \) |
| 29 | \( 1 + (8.54 - 2.36i)T + (24.8 - 14.9i)T^{2} \) |
| 31 | \( 1 + (4.46 - 0.887i)T + (28.6 - 11.8i)T^{2} \) |
| 37 | \( 1 + (1.95 + 8.69i)T + (-33.4 + 15.8i)T^{2} \) |
| 41 | \( 1 + (1.15 - 1.04i)T + (4.01 - 40.8i)T^{2} \) |
| 43 | \( 1 + (0.749 - 1.48i)T + (-25.6 - 34.5i)T^{2} \) |
| 47 | \( 1 + (-7.25 - 3.87i)T + (26.1 + 39.0i)T^{2} \) |
| 53 | \( 1 + (0.00441 + 0.00122i)T + (45.4 + 27.2i)T^{2} \) |
| 59 | \( 1 + (-3.24 + 0.0797i)T + (58.9 - 2.89i)T^{2} \) |
| 61 | \( 1 + (9.03 + 7.79i)T + (8.95 + 60.3i)T^{2} \) |
| 67 | \( 1 + (-10.4 + 0.773i)T + (66.2 - 9.83i)T^{2} \) |
| 71 | \( 1 + (0.433 - 2.91i)T + (-67.9 - 20.6i)T^{2} \) |
| 73 | \( 1 + (3.42 + 5.70i)T + (-34.4 + 64.3i)T^{2} \) |
| 79 | \( 1 + (-14.1 - 4.30i)T + (65.6 + 43.8i)T^{2} \) |
| 83 | \( 1 + (2.07 - 9.22i)T + (-75.0 - 35.4i)T^{2} \) |
| 89 | \( 1 + (0.747 + 1.57i)T + (-56.4 + 68.7i)T^{2} \) |
| 97 | \( 1 + (-9.50 - 6.34i)T + (37.1 + 89.6i)T^{2} \) |
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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
−10.95598092612647255622811959282, −10.69581465861475492391991543544, −9.474614580202344247704963833989, −9.258190989357560332252911649405, −7.51986094357510237442262489945, −7.04277533514288597259215768581, −5.14390496576387817386344780603, −4.28480134733438268511052434196, −3.59581251179648347184138017176, −2.14982716163361219476268642166,
0.40267581498923360060486026718, 1.68257856228275842663212951735, 4.02327432609102319681663784810, 5.36192602548163340223967475891, 5.92518803255073754861050392865, 6.92504258186072667431122403660, 7.969804189215575666666680057777, 8.455039138121711817804853797690, 9.169906786697912357109873631598, 10.59969782030423871651932322706
