| L(s) = 1 | − 2.46·5-s − 7-s − 4.64·11-s + 7.10·13-s + 4.51·17-s + 4.32·19-s − 5.86·23-s + 1.05·25-s + 6.97·29-s − 7.38·31-s + 2.46·35-s − 0.726·37-s − 0.273·41-s − 4.83·43-s − 3.67·47-s + 49-s − 5.05·53-s + 11.4·55-s − 9.13·59-s − 13.8·61-s − 17.4·65-s − 1.32·67-s − 13.5·71-s − 4.32·73-s + 4.64·77-s + 6.43·79-s − 1.48·83-s + ⋯ |
| L(s) = 1 | − 1.10·5-s − 0.377·7-s − 1.40·11-s + 1.97·13-s + 1.09·17-s + 0.992·19-s − 1.22·23-s + 0.210·25-s + 1.29·29-s − 1.32·31-s + 0.415·35-s − 0.119·37-s − 0.0426·41-s − 0.737·43-s − 0.535·47-s + 0.142·49-s − 0.694·53-s + 1.54·55-s − 1.18·59-s − 1.76·61-s − 2.16·65-s − 0.162·67-s − 1.60·71-s − 0.506·73-s + 0.529·77-s + 0.724·79-s − 0.163·83-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 | \( 1 + 2.46T + 5T^{2} \) |
| 11 | \( 1 + 4.64T + 11T^{2} \) |
| 13 | \( 1 - 7.10T + 13T^{2} \) |
| 17 | \( 1 - 4.51T + 17T^{2} \) |
| 19 | \( 1 - 4.32T + 19T^{2} \) |
| 23 | \( 1 + 5.86T + 23T^{2} \) |
| 29 | \( 1 - 6.97T + 29T^{2} \) |
| 31 | \( 1 + 7.38T + 31T^{2} \) |
| 37 | \( 1 + 0.726T + 37T^{2} \) |
| 41 | \( 1 + 0.273T + 41T^{2} \) |
| 43 | \( 1 + 4.83T + 43T^{2} \) |
| 47 | \( 1 + 3.67T + 47T^{2} \) |
| 53 | \( 1 + 5.05T + 53T^{2} \) |
| 59 | \( 1 + 9.13T + 59T^{2} \) |
| 61 | \( 1 + 13.8T + 61T^{2} \) |
| 67 | \( 1 + 1.32T + 67T^{2} \) |
| 71 | \( 1 + 13.5T + 71T^{2} \) |
| 73 | \( 1 + 4.32T + 73T^{2} \) |
| 79 | \( 1 - 6.43T + 79T^{2} \) |
| 83 | \( 1 + 1.48T + 83T^{2} \) |
| 89 | \( 1 + 9.83T + 89T^{2} \) |
| 97 | \( 1 + 0.492T + 97T^{2} \) |
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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
−8.359228266227260414161788887750, −7.956217719277750823018751468049, −7.32398641793239450234749435948, −6.17311469882753718403133241512, −5.57693599452056244082958668793, −4.51783179587490612958701491852, −3.51113502718892552129862641416, −3.09845862426834219052109977794, −1.42655517892776344248016786851, 0,
1.42655517892776344248016786851, 3.09845862426834219052109977794, 3.51113502718892552129862641416, 4.51783179587490612958701491852, 5.57693599452056244082958668793, 6.17311469882753718403133241512, 7.32398641793239450234749435948, 7.956217719277750823018751468049, 8.359228266227260414161788887750
