| L(s) = 1 | − 2·5-s − 7-s − 6·13-s + 8·23-s − 25-s − 6·29-s − 8·31-s + 2·35-s − 10·37-s − 6·41-s − 12·43-s + 49-s + 10·53-s − 8·59-s − 6·61-s + 12·65-s − 12·67-s + 6·73-s − 8·79-s + 16·83-s − 2·89-s + 6·91-s − 2·97-s + 101-s + 103-s + 107-s + 109-s + ⋯ |
| L(s) = 1 | − 0.894·5-s − 0.377·7-s − 1.66·13-s + 1.66·23-s − 1/5·25-s − 1.11·29-s − 1.43·31-s + 0.338·35-s − 1.64·37-s − 0.937·41-s − 1.82·43-s + 1/7·49-s + 1.37·53-s − 1.04·59-s − 0.768·61-s + 1.48·65-s − 1.46·67-s + 0.702·73-s − 0.900·79-s + 1.75·83-s − 0.211·89-s + 0.628·91-s − 0.203·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 291312 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 291312 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 17 | \( 1 \) |
| good | 5 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 12 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 10 T + p T^{2} \) |
| 59 | \( 1 + 8 T + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 - 16 T + p T^{2} \) |
| 89 | \( 1 + 2 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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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
−12.94523512728375, −12.31520756199419, −12.05089396501197, −11.72578261907504, −11.13264327302469, −10.58122362156956, −10.35038650520179, −9.624362840706033, −9.210893785196875, −8.942402415160740, −8.214886623117067, −7.778758907223344, −7.273741256962294, −6.937687871057906, −6.667444151238688, −5.627996128131053, −5.344839016519300, −4.878976246958685, −4.317788480601074, −3.697954528998384, −3.226703739630891, −2.864404256978975, −1.941390071836810, −1.613701948548430, −0.4608432761669463, 0,
0.4608432761669463, 1.613701948548430, 1.941390071836810, 2.864404256978975, 3.226703739630891, 3.697954528998384, 4.317788480601074, 4.878976246958685, 5.344839016519300, 5.627996128131053, 6.667444151238688, 6.937687871057906, 7.273741256962294, 7.778758907223344, 8.214886623117067, 8.942402415160740, 9.210893785196875, 9.624362840706033, 10.35038650520179, 10.58122362156956, 11.13264327302469, 11.72578261907504, 12.05089396501197, 12.31520756199419, 12.94523512728375
