L(s) = 1 | + (−0.866 + 0.5i)2-s + (−0.866 − 0.5i)3-s + (0.499 − 0.866i)4-s + (1.64 + 1.51i)5-s + 0.999·6-s + (1.09 + 0.631i)7-s + 0.999i·8-s + (0.499 + 0.866i)9-s + (−2.18 − 0.494i)10-s − 4.99·11-s + (−0.866 + 0.499i)12-s + (4.64 + 2.68i)13-s − 1.26·14-s + (−0.662 − 2.13i)15-s + (−0.5 − 0.866i)16-s + (−3.69 + 2.13i)17-s + ⋯ |
L(s) = 1 | + (−0.612 + 0.353i)2-s + (−0.499 − 0.288i)3-s + (0.249 − 0.433i)4-s + (0.734 + 0.679i)5-s + 0.408·6-s + (0.413 + 0.238i)7-s + 0.353i·8-s + (0.166 + 0.288i)9-s + (−0.689 − 0.156i)10-s − 1.50·11-s + (−0.249 + 0.144i)12-s + (1.28 + 0.743i)13-s − 0.337·14-s + (−0.171 − 0.551i)15-s + (−0.125 − 0.216i)16-s + (−0.895 + 0.516i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.781 - 0.623i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.781 - 0.623i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.7029083375\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7029083375\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + (0.866 + 0.5i)T \) |
| 5 | \( 1 + (-1.64 - 1.51i)T \) |
| 37 | \( 1 + (-5.93 + 1.32i)T \) |
good | 7 | \( 1 + (-1.09 - 0.631i)T + (3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + 4.99T + 11T^{2} \) |
| 13 | \( 1 + (-4.64 - 2.68i)T + (6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 + (3.69 - 2.13i)T + (8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (3.67 - 6.36i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + 7.94iT - 23T^{2} \) |
| 29 | \( 1 - 0.263T + 29T^{2} \) |
| 31 | \( 1 + 8.35T + 31T^{2} \) |
| 41 | \( 1 + (-3.74 + 6.47i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 - 11.0iT - 43T^{2} \) |
| 47 | \( 1 - 3.66iT - 47T^{2} \) |
| 53 | \( 1 + (7.06 - 4.07i)T + (26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (2.45 + 4.24i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-0.390 + 0.675i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-3.05 - 1.76i)T + (33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (0.141 - 0.244i)T + (-35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 - 1.54iT - 73T^{2} \) |
| 79 | \( 1 + (4.05 - 7.01i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (11.9 - 6.88i)T + (41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + (-6.52 - 11.2i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 - 6.02iT - 97T^{2} \) |
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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
−10.36560991343164038274100830641, −9.309389926847616240064504527423, −8.384914887777768768551025236577, −7.80950654938223024072918239032, −6.61727609404196841481744212672, −6.17207783087529199506753294798, −5.40996068090019756622005048064, −4.17059897026232960500756924552, −2.48808874343640172834337186499, −1.67035458287016544316508328208,
0.38842667956626580711845355556, 1.76832589161977493746644034543, 2.98215428024847378960959808473, 4.38544566166314310024937313408, 5.25438461911662957937555842596, 5.97521844327512785157529078699, 7.15217121782122453245800499339, 8.063985414760829139485016470081, 8.862103708284028487310862383353, 9.499693739107610529628318047347