| L(s) = 1 | − 1.31·5-s − 7-s − 0.418·11-s + 2.27·13-s + 4.77·17-s − 8.54·19-s + 8.71·23-s − 3.27·25-s − 8.24·29-s − 6·31-s + 1.31·35-s + 3.54·37-s + 6.92·41-s + 1.27·43-s − 3.46·47-s + 49-s − 2.20·53-s + 0.549·55-s − 8.71·59-s − 3.72·61-s − 2.98·65-s − 7.27·67-s − 3.88·71-s − 10.2·73-s + 0.418·77-s − 11.2·79-s − 9.55·83-s + ⋯ |
| L(s) = 1 | − 0.587·5-s − 0.377·7-s − 0.126·11-s + 0.630·13-s + 1.15·17-s − 1.96·19-s + 1.81·23-s − 0.654·25-s − 1.53·29-s − 1.07·31-s + 0.222·35-s + 0.583·37-s + 1.08·41-s + 0.194·43-s − 0.505·47-s + 0.142·49-s − 0.303·53-s + 0.0741·55-s − 1.13·59-s − 0.476·61-s − 0.370·65-s − 0.888·67-s − 0.460·71-s − 1.20·73-s + 0.0477·77-s − 1.26·79-s − 1.04·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2268 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2268 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + T \) |
| good | 5 | \( 1 + 1.31T + 5T^{2} \) |
| 11 | \( 1 + 0.418T + 11T^{2} \) |
| 13 | \( 1 - 2.27T + 13T^{2} \) |
| 17 | \( 1 - 4.77T + 17T^{2} \) |
| 19 | \( 1 + 8.54T + 19T^{2} \) |
| 23 | \( 1 - 8.71T + 23T^{2} \) |
| 29 | \( 1 + 8.24T + 29T^{2} \) |
| 31 | \( 1 + 6T + 31T^{2} \) |
| 37 | \( 1 - 3.54T + 37T^{2} \) |
| 41 | \( 1 - 6.92T + 41T^{2} \) |
| 43 | \( 1 - 1.27T + 43T^{2} \) |
| 47 | \( 1 + 3.46T + 47T^{2} \) |
| 53 | \( 1 + 2.20T + 53T^{2} \) |
| 59 | \( 1 + 8.71T + 59T^{2} \) |
| 61 | \( 1 + 3.72T + 61T^{2} \) |
| 67 | \( 1 + 7.27T + 67T^{2} \) |
| 71 | \( 1 + 3.88T + 71T^{2} \) |
| 73 | \( 1 + 10.2T + 73T^{2} \) |
| 79 | \( 1 + 11.2T + 79T^{2} \) |
| 83 | \( 1 + 9.55T + 83T^{2} \) |
| 89 | \( 1 - 2.98T + 89T^{2} \) |
| 97 | \( 1 + 0.549T + 97T^{2} \) |
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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
−8.727061964136946782660105745292, −7.75698567249683280770613442357, −7.25826272370987891686537377895, −6.21498846718421279853388539139, −5.60429618561184585105620631640, −4.45444040093233669566429807323, −3.72648633435500483013278207585, −2.85971322652693771671091939998, −1.52286999484318224953156369478, 0,
1.52286999484318224953156369478, 2.85971322652693771671091939998, 3.72648633435500483013278207585, 4.45444040093233669566429807323, 5.60429618561184585105620631640, 6.21498846718421279853388539139, 7.25826272370987891686537377895, 7.75698567249683280770613442357, 8.727061964136946782660105745292
