L(s) = 1 | + (−0.834 + 1.44i)2-s + (−1.40 − 2.43i)3-s + (−0.393 − 0.681i)4-s + (1.40 − 2.43i)5-s + 4.69·6-s + (−2 − 1.73i)7-s − 2.02·8-s + (−2.45 + 4.25i)9-s + (2.34 + 4.06i)10-s + (0.5 + 0.866i)11-s + (−1.10 + 1.91i)12-s − 13-s + (4.17 − 1.44i)14-s − 7.90·15-s + (2.47 − 4.29i)16-s + (−2.51 − 4.35i)17-s + ⋯ |
L(s) = 1 | + (−0.590 + 1.02i)2-s + (−0.811 − 1.40i)3-s + (−0.196 − 0.340i)4-s + (0.628 − 1.08i)5-s + 1.91·6-s + (−0.755 − 0.654i)7-s − 0.715·8-s + (−0.817 + 1.41i)9-s + (0.742 + 1.28i)10-s + (0.150 + 0.261i)11-s + (−0.319 + 0.553i)12-s − 0.277·13-s + (1.11 − 0.386i)14-s − 2.04·15-s + (0.619 − 1.07i)16-s + (−0.609 − 1.05i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1001 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.991 - 0.126i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1001 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.991 - 0.126i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.2639519970\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.2639519970\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 7 | \( 1 + (2 + 1.73i)T \) |
| 11 | \( 1 + (-0.5 - 0.866i)T \) |
| 13 | \( 1 + T \) |
good | 2 | \( 1 + (0.834 - 1.44i)T + (-1 - 1.73i)T^{2} \) |
| 3 | \( 1 + (1.40 + 2.43i)T + (-1.5 + 2.59i)T^{2} \) |
| 5 | \( 1 + (-1.40 + 2.43i)T + (-2.5 - 4.33i)T^{2} \) |
| 17 | \( 1 + (2.51 + 4.35i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-2.69 + 4.65i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (0.271 - 0.470i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + 0.974T + 29T^{2} \) |
| 31 | \( 1 + (-1.02 - 1.77i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (1.02 - 1.77i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 - 0.355T + 41T^{2} \) |
| 43 | \( 1 + 10.0T + 43T^{2} \) |
| 47 | \( 1 + (0.441 - 0.764i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (-4.53 - 7.86i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (-4.83 - 8.37i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (4.13 - 7.16i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (4.14 + 7.17i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + 14.0T + 71T^{2} \) |
| 73 | \( 1 + (-5.56 - 9.63i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (-5.97 + 10.3i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + 12.2T + 83T^{2} \) |
| 89 | \( 1 + (-0.327 + 0.567i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 - 0.952T + 97T^{2} \) |
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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
−9.247938663845343353645552994475, −8.643505179169878911333119464334, −7.40755895295894992751759815507, −7.14491938056734749548271163502, −6.38474340615680388320300146420, −5.58961738338954304210590609001, −4.75099745917082462340362015345, −2.77972782441176388148755721982, −1.22471206379033969766324618253, −0.17257459404037613051318386196,
2.03272947816900124217668441931, 3.15795965127436589787455842325, 3.82042591414134241989307281571, 5.35654303322204670661153080213, 6.06570735493139220036559800753, 6.61528950561270549485849100728, 8.403942073681504982131578243280, 9.321788337990269209445981966197, 9.950326735028331503728078929652, 10.25668825738115936049466268328