L(s) = 1 | + (0.5 − 0.866i)2-s + (0.866 − 0.5i)3-s + (−0.499 − 0.866i)4-s + (1.50 − 1.65i)5-s − 0.999i·6-s + (−0.827 + 0.478i)7-s − 0.999·8-s + (0.499 − 0.866i)9-s + (−0.682 − 2.12i)10-s + 0.252·11-s + (−0.866 − 0.499i)12-s + (0.858 + 1.48i)13-s + 0.956i·14-s + (0.473 − 2.18i)15-s + (−0.5 + 0.866i)16-s + (2.88 − 4.99i)17-s + ⋯ |
L(s) = 1 | + (0.353 − 0.612i)2-s + (0.499 − 0.288i)3-s + (−0.249 − 0.433i)4-s + (0.671 − 0.740i)5-s − 0.408i·6-s + (−0.312 + 0.180i)7-s − 0.353·8-s + (0.166 − 0.288i)9-s + (−0.215 − 0.673i)10-s + 0.0762·11-s + (−0.249 − 0.144i)12-s + (0.238 + 0.412i)13-s + 0.255i·14-s + (0.122 − 0.564i)15-s + (−0.125 + 0.216i)16-s + (0.698 − 1.21i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.513 + 0.857i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.513 + 0.857i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.411432623\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.411432623\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.5 + 0.866i)T \) |
| 3 | \( 1 + (-0.866 + 0.5i)T \) |
| 5 | \( 1 + (-1.50 + 1.65i)T \) |
| 37 | \( 1 + (3.20 - 5.16i)T \) |
good | 7 | \( 1 + (0.827 - 0.478i)T + (3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 - 0.252T + 11T^{2} \) |
| 13 | \( 1 + (-0.858 - 1.48i)T + (-6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 + (-2.88 + 4.99i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-3.89 + 2.24i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + 4.71T + 23T^{2} \) |
| 29 | \( 1 + 6.74iT - 29T^{2} \) |
| 31 | \( 1 - 0.279iT - 31T^{2} \) |
| 41 | \( 1 + (2.96 + 5.13i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 - 1.27T + 43T^{2} \) |
| 47 | \( 1 - 6.17iT - 47T^{2} \) |
| 53 | \( 1 + (7.10 + 4.10i)T + (26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (-13.0 - 7.52i)T + (29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-3.43 + 1.98i)T + (30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (0.518 - 0.299i)T + (33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (-6.17 - 10.6i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 - 10.7iT - 73T^{2} \) |
| 79 | \( 1 + (2.20 - 1.27i)T + (39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (12.5 + 7.23i)T + (41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + (-6.84 - 3.95i)T + (44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 - 17.8T + 97T^{2} \) |
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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
−9.708950380858226369533515885675, −8.948206493078990860470911012617, −8.100855785573109049372650885670, −7.01859590816614931870308329418, −6.00443101390937148137847332876, −5.21182371444366908851784326235, −4.23421799036176822596412625400, −3.08420477776121819531128979972, −2.14079410941128626254829994569, −0.936645442137941568413577194737,
1.82739458578989805809751399574, 3.28288410820823961998598460481, 3.70294503229561719446029698323, 5.16919032889327396831723200376, 5.91424696628807618449539698726, 6.70972664719274048563417909349, 7.62422529335408132603851404743, 8.329236317381566892373735986372, 9.322767107773799745459775019178, 10.11435094115459822044723717634