| L(s) = 1 | + 2·5-s − 4·11-s + 2·13-s + 4·19-s − 25-s + 10·29-s + 8·31-s − 2·37-s + 10·41-s + 12·43-s − 7·49-s + 6·53-s − 8·55-s − 12·59-s − 10·61-s + 4·65-s − 12·67-s − 10·73-s − 8·79-s − 4·83-s + 6·89-s + 8·95-s + 14·97-s + 101-s + 103-s + 107-s + 109-s + ⋯ |
| L(s) = 1 | + 0.894·5-s − 1.20·11-s + 0.554·13-s + 0.917·19-s − 1/5·25-s + 1.85·29-s + 1.43·31-s − 0.328·37-s + 1.56·41-s + 1.82·43-s − 49-s + 0.824·53-s − 1.07·55-s − 1.56·59-s − 1.28·61-s + 0.496·65-s − 1.46·67-s − 1.17·73-s − 0.900·79-s − 0.439·83-s + 0.635·89-s + 0.820·95-s + 1.42·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 166464 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 166464 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 17 | \( 1 \) |
| good | 5 | \( 1 - 2 T + p T^{2} \) |
| 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 10 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 - 12 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 - 14 T + p T^{2} \) |
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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
−13.51073809924615, −13.14520767817897, −12.52421076358561, −12.11878362959867, −11.64505227358907, −10.94143157334989, −10.59787187936416, −10.10023836466405, −9.839366641819433, −9.075692750607671, −8.848052050279312, −8.109906440403788, −7.642989127294953, −7.357290805427730, −6.396039501846678, −6.126061621362161, −5.738429908750453, −5.059984516347296, −4.617433066693001, −4.093929925539574, −3.130539300814295, −2.763464914366432, −2.368118854087426, −1.382891920373173, −1.022530581801035, 0,
1.022530581801035, 1.382891920373173, 2.368118854087426, 2.763464914366432, 3.130539300814295, 4.093929925539574, 4.617433066693001, 5.059984516347296, 5.738429908750453, 6.126061621362161, 6.396039501846678, 7.357290805427730, 7.642989127294953, 8.109906440403788, 8.848052050279312, 9.075692750607671, 9.839366641819433, 10.10023836466405, 10.59787187936416, 10.94143157334989, 11.64505227358907, 12.11878362959867, 12.52421076358561, 13.14520767817897, 13.51073809924615
