| L(s) = 1 | + 2.23·3-s − 2.61·7-s + 2.00·9-s − 1.23·11-s + 3.38·13-s + 5.23·17-s − 7.61·19-s − 5.85·21-s − 5.47·23-s − 2.23·27-s + 4.38·29-s + 4.70·31-s − 2.76·33-s − 8.23·37-s + 7.56·39-s + 5.70·41-s − 4.38·43-s − 0.145·47-s − 0.145·49-s + 11.7·51-s + 7.94·53-s − 17.0·57-s − 2.38·59-s − 13.1·61-s − 5.23·63-s − 13.7·67-s − 12.2·69-s + ⋯ |
| L(s) = 1 | + 1.29·3-s − 0.989·7-s + 0.666·9-s − 0.372·11-s + 0.937·13-s + 1.26·17-s − 1.74·19-s − 1.27·21-s − 1.14·23-s − 0.430·27-s + 0.813·29-s + 0.845·31-s − 0.481·33-s − 1.35·37-s + 1.21·39-s + 0.891·41-s − 0.668·43-s − 0.0212·47-s − 0.0208·49-s + 1.63·51-s + 1.09·53-s − 2.25·57-s − 0.310·59-s − 1.68·61-s − 0.659·63-s − 1.67·67-s − 1.47·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 10000 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 10000 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 \) |
| good | 3 | \( 1 - 2.23T + 3T^{2} \) |
| 7 | \( 1 + 2.61T + 7T^{2} \) |
| 11 | \( 1 + 1.23T + 11T^{2} \) |
| 13 | \( 1 - 3.38T + 13T^{2} \) |
| 17 | \( 1 - 5.23T + 17T^{2} \) |
| 19 | \( 1 + 7.61T + 19T^{2} \) |
| 23 | \( 1 + 5.47T + 23T^{2} \) |
| 29 | \( 1 - 4.38T + 29T^{2} \) |
| 31 | \( 1 - 4.70T + 31T^{2} \) |
| 37 | \( 1 + 8.23T + 37T^{2} \) |
| 41 | \( 1 - 5.70T + 41T^{2} \) |
| 43 | \( 1 + 4.38T + 43T^{2} \) |
| 47 | \( 1 + 0.145T + 47T^{2} \) |
| 53 | \( 1 - 7.94T + 53T^{2} \) |
| 59 | \( 1 + 2.38T + 59T^{2} \) |
| 61 | \( 1 + 13.1T + 61T^{2} \) |
| 67 | \( 1 + 13.7T + 67T^{2} \) |
| 71 | \( 1 - 8.61T + 71T^{2} \) |
| 73 | \( 1 - 11T + 73T^{2} \) |
| 79 | \( 1 - 5.32T + 79T^{2} \) |
| 83 | \( 1 - 4.52T + 83T^{2} \) |
| 89 | \( 1 - 4T + 89T^{2} \) |
| 97 | \( 1 + 15.2T + 97T^{2} \) |
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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
−7.52067240281180153504138996552, −6.51262923171420901196638422480, −6.20391333795979263369329483070, −5.30566257371263762894874685016, −4.19112495100410277533944944911, −3.68850670177880344444030869434, −3.00846747387846470259935183423, −2.38288952277602740832530962922, −1.41929803902827497715119417380, 0,
1.41929803902827497715119417380, 2.38288952277602740832530962922, 3.00846747387846470259935183423, 3.68850670177880344444030869434, 4.19112495100410277533944944911, 5.30566257371263762894874685016, 6.20391333795979263369329483070, 6.51262923171420901196638422480, 7.52067240281180153504138996552
