| L(s) = 1 | + 6.30·5-s − 7·7-s + 48.9·11-s − 2.60·13-s − 136.·17-s − 45.2·19-s − 38.1·23-s − 85.2·25-s − 52.7·29-s + 14.7·31-s − 44.1·35-s + 333.·37-s − 227.·41-s + 398.·43-s − 184.·47-s + 49·49-s − 359.·53-s + 308.·55-s + 99.9·59-s − 674.·61-s − 16.4·65-s + 376.·67-s − 1.18e3·71-s − 735.·73-s − 342.·77-s + 836.·79-s + 293.·83-s + ⋯ |
| L(s) = 1 | + 0.563·5-s − 0.377·7-s + 1.34·11-s − 0.0556·13-s − 1.95·17-s − 0.545·19-s − 0.345·23-s − 0.682·25-s − 0.337·29-s + 0.0856·31-s − 0.213·35-s + 1.48·37-s − 0.865·41-s + 1.41·43-s − 0.572·47-s + 0.142·49-s − 0.932·53-s + 0.755·55-s + 0.220·59-s − 1.41·61-s − 0.0313·65-s + 0.687·67-s − 1.98·71-s − 1.17·73-s − 0.506·77-s + 1.19·79-s + 0.388·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + 7T \) |
| good | 5 | \( 1 - 6.30T + 125T^{2} \) |
| 11 | \( 1 - 48.9T + 1.33e3T^{2} \) |
| 13 | \( 1 + 2.60T + 2.19e3T^{2} \) |
| 17 | \( 1 + 136.T + 4.91e3T^{2} \) |
| 19 | \( 1 + 45.2T + 6.85e3T^{2} \) |
| 23 | \( 1 + 38.1T + 1.21e4T^{2} \) |
| 29 | \( 1 + 52.7T + 2.43e4T^{2} \) |
| 31 | \( 1 - 14.7T + 2.97e4T^{2} \) |
| 37 | \( 1 - 333.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 227.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 398.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 184.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 359.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 99.9T + 2.05e5T^{2} \) |
| 61 | \( 1 + 674.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 376.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 1.18e3T + 3.57e5T^{2} \) |
| 73 | \( 1 + 735.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 836.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 293.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 1.29e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 201.T + 9.12e5T^{2} \) |
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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
−9.236378989940766534048141790688, −8.557497873834533495646956044863, −7.36882942875971443385206816302, −6.39188239121309759532992275516, −6.06159912957103008621050021437, −4.60665405853406483759528302819, −3.90997272487971659521127227672, −2.53980732672272079458197846216, −1.55937563887268020883397695451, 0,
1.55937563887268020883397695451, 2.53980732672272079458197846216, 3.90997272487971659521127227672, 4.60665405853406483759528302819, 6.06159912957103008621050021437, 6.39188239121309759532992275516, 7.36882942875971443385206816302, 8.557497873834533495646956044863, 9.236378989940766534048141790688
