| L(s) = 1 | + 2.43·2-s + 3·3-s − 2.09·4-s + 1.55·5-s + 7.29·6-s + 20.4·7-s − 24.5·8-s + 9·9-s + 3.78·10-s + 11·11-s − 6.28·12-s + 30.9·13-s + 49.7·14-s + 4.67·15-s − 42.8·16-s − 56.6·17-s + 21.8·18-s − 135.·19-s − 3.26·20-s + 61.4·21-s + 26.7·22-s − 175.·23-s − 73.5·24-s − 122.·25-s + 75.2·26-s + 27·27-s − 42.8·28-s + ⋯ |
| L(s) = 1 | + 0.859·2-s + 0.577·3-s − 0.261·4-s + 0.139·5-s + 0.496·6-s + 1.10·7-s − 1.08·8-s + 0.333·9-s + 0.119·10-s + 0.301·11-s − 0.151·12-s + 0.660·13-s + 0.949·14-s + 0.0805·15-s − 0.669·16-s − 0.808·17-s + 0.286·18-s − 1.63·19-s − 0.0365·20-s + 0.638·21-s + 0.259·22-s − 1.59·23-s − 0.625·24-s − 0.980·25-s + 0.567·26-s + 0.192·27-s − 0.289·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2013 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2013 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - 3T \) |
| 11 | \( 1 - 11T \) |
| 61 | \( 1 - 61T \) |
| good | 2 | \( 1 - 2.43T + 8T^{2} \) |
| 5 | \( 1 - 1.55T + 125T^{2} \) |
| 7 | \( 1 - 20.4T + 343T^{2} \) |
| 13 | \( 1 - 30.9T + 2.19e3T^{2} \) |
| 17 | \( 1 + 56.6T + 4.91e3T^{2} \) |
| 19 | \( 1 + 135.T + 6.85e3T^{2} \) |
| 23 | \( 1 + 175.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 88.0T + 2.43e4T^{2} \) |
| 31 | \( 1 - 213.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 200.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 161.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 276.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 139.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 362.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 587.T + 2.05e5T^{2} \) |
| 67 | \( 1 + 417.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 722.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 532.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 696.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 603.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 228.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.65e3T + 9.12e5T^{2} \) |
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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
−8.539552899371742439078965990705, −7.83664404694201172682080553444, −6.51895186055280643584630148480, −6.03525373855482302727639998781, −4.89037468872476127777281933332, −4.29140064778109179016596090997, −3.68814172487578546332373324864, −2.42427748956070910349300529727, −1.61715576578397146856702731827, 0,
1.61715576578397146856702731827, 2.42427748956070910349300529727, 3.68814172487578546332373324864, 4.29140064778109179016596090997, 4.89037468872476127777281933332, 6.03525373855482302727639998781, 6.51895186055280643584630148480, 7.83664404694201172682080553444, 8.539552899371742439078965990705
