L(s) = 1 | + 2-s − 3-s + 4-s − 3.15·5-s − 6-s − 5.11·7-s + 8-s + 9-s − 3.15·10-s + 2.30·11-s − 12-s − 0.811·13-s − 5.11·14-s + 3.15·15-s + 16-s + 17-s + 18-s − 5.24·19-s − 3.15·20-s + 5.11·21-s + 2.30·22-s − 3.26·23-s − 24-s + 4.94·25-s − 0.811·26-s − 27-s − 5.11·28-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 0.577·3-s + 0.5·4-s − 1.41·5-s − 0.408·6-s − 1.93·7-s + 0.353·8-s + 0.333·9-s − 0.997·10-s + 0.694·11-s − 0.288·12-s − 0.224·13-s − 1.36·14-s + 0.814·15-s + 0.250·16-s + 0.242·17-s + 0.235·18-s − 1.20·19-s − 0.705·20-s + 1.11·21-s + 0.490·22-s − 0.681·23-s − 0.204·24-s + 0.988·25-s − 0.159·26-s − 0.192·27-s − 0.967·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6018 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6018 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.4966120274\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.4966120274\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T \) |
| 3 | \( 1 + T \) |
| 17 | \( 1 - T \) |
| 59 | \( 1 - T \) |
good | 5 | \( 1 + 3.15T + 5T^{2} \) |
| 7 | \( 1 + 5.11T + 7T^{2} \) |
| 11 | \( 1 - 2.30T + 11T^{2} \) |
| 13 | \( 1 + 0.811T + 13T^{2} \) |
| 19 | \( 1 + 5.24T + 19T^{2} \) |
| 23 | \( 1 + 3.26T + 23T^{2} \) |
| 29 | \( 1 + 8.27T + 29T^{2} \) |
| 31 | \( 1 + 8.46T + 31T^{2} \) |
| 37 | \( 1 + 4.64T + 37T^{2} \) |
| 41 | \( 1 + 4.54T + 41T^{2} \) |
| 43 | \( 1 - 0.284T + 43T^{2} \) |
| 47 | \( 1 + 6.61T + 47T^{2} \) |
| 53 | \( 1 - 13.2T + 53T^{2} \) |
| 61 | \( 1 + 2.14T + 61T^{2} \) |
| 67 | \( 1 + 13.5T + 67T^{2} \) |
| 71 | \( 1 + 2.05T + 71T^{2} \) |
| 73 | \( 1 - 12.6T + 73T^{2} \) |
| 79 | \( 1 - 4.35T + 79T^{2} \) |
| 83 | \( 1 - 7.85T + 83T^{2} \) |
| 89 | \( 1 - 5.97T + 89T^{2} \) |
| 97 | \( 1 - 5.62T + 97T^{2} \) |
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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
−7.79461719229342691023707874999, −7.11432029082803100557609425000, −6.66040807003015890400734991297, −5.99862769081071472100204307422, −5.26209068404624883602253197348, −4.08778563350214521071427873642, −3.82107109058613997143617108207, −3.22163835568067705151474835263, −1.97725464477876745243594003818, −0.32182456169688634505070105854,
0.32182456169688634505070105854, 1.97725464477876745243594003818, 3.22163835568067705151474835263, 3.82107109058613997143617108207, 4.08778563350214521071427873642, 5.26209068404624883602253197348, 5.99862769081071472100204307422, 6.66040807003015890400734991297, 7.11432029082803100557609425000, 7.79461719229342691023707874999