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
−11.15782124276602402605719970348, −10.03269174634094159244269071115, −9.318324256468968088408510378671, −7.69142391746753347386998246056, −6.91557202736186785793798584034, −6.29043475376490839500308371656, −5.33227071454830488358601285270, −3.58100304210979593434574221023, −3.14827916740683880830491384810, −1.45735749955846818859094979571,
1.26226376673888705563999383917, 3.40070455090560381716613569849, 4.31654837945409076956688125335, 5.26073625750644198636961068661, 5.89574126155553846202212122519, 7.09334399883042678238062887757, 8.331701478368411641822987243923, 9.009074783079621196302698514583, 9.790884110916774934292165792518, 10.98435885028860441369972802543