L(s) = 1 | + (0.866 + 0.5i)2-s + (−0.5 + 0.866i)3-s + (0.499 + 0.866i)4-s − 3.46i·5-s + (−0.866 + 0.499i)6-s + (−0.866 + 0.5i)7-s + 0.999i·8-s + (−0.499 − 0.866i)9-s + (1.73 − 2.99i)10-s + (3.46 + 2i)11-s − 0.999·12-s + (2.59 − 2.5i)13-s − 0.999·14-s + (2.99 + 1.73i)15-s + (−0.5 + 0.866i)16-s + (3.23 + 5.59i)17-s + ⋯ |
L(s) = 1 | + (0.612 + 0.353i)2-s + (−0.288 + 0.499i)3-s + (0.249 + 0.433i)4-s − 1.54i·5-s + (−0.353 + 0.204i)6-s + (−0.327 + 0.188i)7-s + 0.353i·8-s + (−0.166 − 0.288i)9-s + (0.547 − 0.948i)10-s + (1.04 + 0.603i)11-s − 0.288·12-s + (0.720 − 0.693i)13-s − 0.267·14-s + (0.774 + 0.447i)15-s + (−0.125 + 0.216i)16-s + (0.783 + 1.35i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.967 - 0.252i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.967 - 0.252i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.95640 + 0.251186i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.95640 + 0.251186i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.866 - 0.5i)T \) |
| 3 | \( 1 + (0.5 - 0.866i)T \) |
| 7 | \( 1 + (0.866 - 0.5i)T \) |
| 13 | \( 1 + (-2.59 + 2.5i)T \) |
good | 5 | \( 1 + 3.46iT - 5T^{2} \) |
| 11 | \( 1 + (-3.46 - 2i)T + (5.5 + 9.52i)T^{2} \) |
| 17 | \( 1 + (-3.23 - 5.59i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-6.46 + 3.73i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-2.86 + 4.96i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (4 - 6.92i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + 6.46iT - 31T^{2} \) |
| 37 | \( 1 + (4.26 + 2.46i)T + (18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (6 + 3.46i)T + (20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (-2.23 - 3.86i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 - 0.535iT - 47T^{2} \) |
| 53 | \( 1 + 8.26T + 53T^{2} \) |
| 59 | \( 1 + (5.42 - 3.13i)T + (29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-2.59 - 4.5i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-6.23 - 3.59i)T + (33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (-1.66 + 0.964i)T + (35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 - 10.9iT - 73T^{2} \) |
| 79 | \( 1 + 9.46T + 79T^{2} \) |
| 83 | \( 1 - 1.73iT - 83T^{2} \) |
| 89 | \( 1 + (10.1 + 5.86i)T + (44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (-10.7 + 6.19i)T + (48.5 - 84.0i)T^{2} \) |
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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
−10.98435885028860441369972802543, −9.790884110916774934292165792518, −9.009074783079621196302698514583, −8.331701478368411641822987243923, −7.09334399883042678238062887757, −5.89574126155553846202212122519, −5.26073625750644198636961068661, −4.31654837945409076956688125335, −3.40070455090560381716613569849, −1.26226376673888705563999383917,
1.45735749955846818859094979571, 3.14827916740683880830491384810, 3.58100304210979593434574221023, 5.33227071454830488358601285270, 6.29043475376490839500308371656, 6.91557202736186785793798584034, 7.69142391746753347386998246056, 9.318324256468968088408510378671, 10.03269174634094159244269071115, 11.15782124276602402605719970348