| L(s) = 1 | + 1.23·3-s − 0.600·5-s − 7-s − 1.48·9-s − 11-s + 13-s − 0.739·15-s + 1.80·17-s + 4.27·19-s − 1.23·21-s + 3.07·23-s − 4.63·25-s − 5.52·27-s − 1.52·29-s − 6.09·31-s − 1.23·33-s + 0.600·35-s + 0.452·37-s + 1.23·39-s − 5.59·41-s − 5.23·43-s + 0.890·45-s − 7.70·47-s + 49-s + 2.22·51-s + 2.83·53-s + 0.600·55-s + ⋯ |
| L(s) = 1 | + 0.711·3-s − 0.268·5-s − 0.377·7-s − 0.494·9-s − 0.301·11-s + 0.277·13-s − 0.190·15-s + 0.438·17-s + 0.980·19-s − 0.268·21-s + 0.640·23-s − 0.927·25-s − 1.06·27-s − 0.282·29-s − 1.09·31-s − 0.214·33-s + 0.101·35-s + 0.0743·37-s + 0.197·39-s − 0.873·41-s − 0.798·43-s + 0.132·45-s − 1.12·47-s + 0.142·49-s + 0.311·51-s + 0.389·53-s + 0.0809·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4004 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4004 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 + T \) |
| 13 | \( 1 - T \) |
| good | 3 | \( 1 - 1.23T + 3T^{2} \) |
| 5 | \( 1 + 0.600T + 5T^{2} \) |
| 17 | \( 1 - 1.80T + 17T^{2} \) |
| 19 | \( 1 - 4.27T + 19T^{2} \) |
| 23 | \( 1 - 3.07T + 23T^{2} \) |
| 29 | \( 1 + 1.52T + 29T^{2} \) |
| 31 | \( 1 + 6.09T + 31T^{2} \) |
| 37 | \( 1 - 0.452T + 37T^{2} \) |
| 41 | \( 1 + 5.59T + 41T^{2} \) |
| 43 | \( 1 + 5.23T + 43T^{2} \) |
| 47 | \( 1 + 7.70T + 47T^{2} \) |
| 53 | \( 1 - 2.83T + 53T^{2} \) |
| 59 | \( 1 - 7.62T + 59T^{2} \) |
| 61 | \( 1 - 6.76T + 61T^{2} \) |
| 67 | \( 1 - 7.57T + 67T^{2} \) |
| 71 | \( 1 + 4.36T + 71T^{2} \) |
| 73 | \( 1 + 1.56T + 73T^{2} \) |
| 79 | \( 1 + 11.4T + 79T^{2} \) |
| 83 | \( 1 + 9.00T + 83T^{2} \) |
| 89 | \( 1 + 15.9T + 89T^{2} \) |
| 97 | \( 1 + 9.57T + 97T^{2} \) |
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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
−8.228982821120442747592130284206, −7.41562300978532681714545033797, −6.80615802323738318093285054963, −5.67249912162230661670790912096, −5.27656228687718838224878237527, −3.97565884692163527959756667806, −3.36478278667805561269729983697, −2.67123439391022422331579647085, −1.52796691061427951186904519525, 0,
1.52796691061427951186904519525, 2.67123439391022422331579647085, 3.36478278667805561269729983697, 3.97565884692163527959756667806, 5.27656228687718838224878237527, 5.67249912162230661670790912096, 6.80615802323738318093285054963, 7.41562300978532681714545033797, 8.228982821120442747592130284206
