| L(s) = 1 | + (−0.0238 − 1.41i)2-s + (−0.306 − 1.73i)3-s + (−1.99 + 0.0675i)4-s + (−0.220 + 0.184i)5-s + (−2.45 + 0.475i)6-s + (0.588 + 0.339i)7-s + (0.143 + 2.82i)8-s + (−0.110 + 0.0403i)9-s + (0.266 + 0.306i)10-s + (3.85 − 2.22i)11-s + (0.730 + 3.45i)12-s + (−3.41 − 0.602i)13-s + (0.466 − 0.840i)14-s + (0.388 + 0.326i)15-s + (3.99 − 0.270i)16-s + (4.15 + 1.51i)17-s + ⋯ |
| L(s) = 1 | + (−0.0168 − 0.999i)2-s + (−0.177 − 1.00i)3-s + (−0.999 + 0.0337i)4-s + (−0.0984 + 0.0826i)5-s + (−1.00 + 0.193i)6-s + (0.222 + 0.128i)7-s + (0.0506 + 0.998i)8-s + (−0.0369 + 0.0134i)9-s + (0.0842 + 0.0970i)10-s + (1.16 − 0.670i)11-s + (0.210 + 0.997i)12-s + (−0.947 − 0.166i)13-s + (0.124 − 0.224i)14-s + (0.100 + 0.0842i)15-s + (0.997 − 0.0675i)16-s + (1.00 + 0.366i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.481 + 0.876i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.481 + 0.876i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.428108 - 0.723909i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.428108 - 0.723909i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.0238 + 1.41i)T \) |
| 19 | \( 1 + (1.76 - 3.98i)T \) |
| good | 3 | \( 1 + (0.306 + 1.73i)T + (-2.81 + 1.02i)T^{2} \) |
| 5 | \( 1 + (0.220 - 0.184i)T + (0.868 - 4.92i)T^{2} \) |
| 7 | \( 1 + (-0.588 - 0.339i)T + (3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (-3.85 + 2.22i)T + (5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (3.41 + 0.602i)T + (12.2 + 4.44i)T^{2} \) |
| 17 | \( 1 + (-4.15 - 1.51i)T + (13.0 + 10.9i)T^{2} \) |
| 23 | \( 1 + (0.347 - 0.413i)T + (-3.99 - 22.6i)T^{2} \) |
| 29 | \( 1 + (-1.03 - 2.85i)T + (-22.2 + 18.6i)T^{2} \) |
| 31 | \( 1 + (5.24 - 9.08i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + 8.82iT - 37T^{2} \) |
| 41 | \( 1 + (-1.85 + 0.326i)T + (38.5 - 14.0i)T^{2} \) |
| 43 | \( 1 + (3.49 + 4.16i)T + (-7.46 + 42.3i)T^{2} \) |
| 47 | \( 1 + (0.419 + 1.15i)T + (-36.0 + 30.2i)T^{2} \) |
| 53 | \( 1 + (6.41 - 7.64i)T + (-9.20 - 52.1i)T^{2} \) |
| 59 | \( 1 + (-4.20 - 1.53i)T + (45.1 + 37.9i)T^{2} \) |
| 61 | \( 1 + (-6.04 - 5.07i)T + (10.5 + 60.0i)T^{2} \) |
| 67 | \( 1 + (-4.66 + 1.69i)T + (51.3 - 43.0i)T^{2} \) |
| 71 | \( 1 + (-6.81 + 5.72i)T + (12.3 - 69.9i)T^{2} \) |
| 73 | \( 1 + (-0.591 - 3.35i)T + (-68.5 + 24.9i)T^{2} \) |
| 79 | \( 1 + (-1.43 - 8.15i)T + (-74.2 + 27.0i)T^{2} \) |
| 83 | \( 1 + (8.64 + 4.99i)T + (41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + (10.6 + 1.87i)T + (83.6 + 30.4i)T^{2} \) |
| 97 | \( 1 + (1.24 - 3.40i)T + (-74.3 - 62.3i)T^{2} \) |
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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
−14.02119626517136675207708228943, −12.60291725289286122577605466559, −12.23158396072873783966811619881, −11.10108103677465377820961992190, −9.813806174044964910925976625963, −8.520376965713788264721548021130, −7.22947395669401901726018188686, −5.58737017092271667078729815019, −3.65054189296029366959591427245, −1.56380952101430898256648814964,
4.12826679791545416512166739710, 4.98487103009129182876384471820, 6.63749240959283523245845164483, 7.88331497042823728506483223749, 9.449149842906507641370329630240, 9.920090082700821629189304421466, 11.58488113492748735785934248400, 12.87152380088899514267567562119, 14.33727974442257449906474814013, 14.91810791010423193636328732645
