| L(s) = 1 | + 5-s + 1.56·7-s − 3.12·11-s + 2·13-s − 0.438·17-s − 7.12·19-s + 23-s + 25-s − 4.43·29-s + 8.68·31-s + 1.56·35-s − 3.56·37-s − 7.56·41-s + 10.2·43-s − 8·47-s − 4.56·49-s + 3.56·53-s − 3.12·55-s − 2.43·59-s − 11.3·61-s + 2·65-s + 1.56·67-s + 0.684·71-s + 2·73-s − 4.87·77-s − 6.24·79-s + 12.6·83-s + ⋯ |
| L(s) = 1 | + 0.447·5-s + 0.590·7-s − 0.941·11-s + 0.554·13-s − 0.106·17-s − 1.63·19-s + 0.208·23-s + 0.200·25-s − 0.824·29-s + 1.55·31-s + 0.263·35-s − 0.585·37-s − 1.18·41-s + 1.56·43-s − 1.16·47-s − 0.651·49-s + 0.489·53-s − 0.421·55-s − 0.317·59-s − 1.45·61-s + 0.248·65-s + 0.190·67-s + 0.0812·71-s + 0.234·73-s − 0.555·77-s − 0.702·79-s + 1.39·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 - T \) |
| 23 | \( 1 - T \) |
| good | 7 | \( 1 - 1.56T + 7T^{2} \) |
| 11 | \( 1 + 3.12T + 11T^{2} \) |
| 13 | \( 1 - 2T + 13T^{2} \) |
| 17 | \( 1 + 0.438T + 17T^{2} \) |
| 19 | \( 1 + 7.12T + 19T^{2} \) |
| 29 | \( 1 + 4.43T + 29T^{2} \) |
| 31 | \( 1 - 8.68T + 31T^{2} \) |
| 37 | \( 1 + 3.56T + 37T^{2} \) |
| 41 | \( 1 + 7.56T + 41T^{2} \) |
| 43 | \( 1 - 10.2T + 43T^{2} \) |
| 47 | \( 1 + 8T + 47T^{2} \) |
| 53 | \( 1 - 3.56T + 53T^{2} \) |
| 59 | \( 1 + 2.43T + 59T^{2} \) |
| 61 | \( 1 + 11.3T + 61T^{2} \) |
| 67 | \( 1 - 1.56T + 67T^{2} \) |
| 71 | \( 1 - 0.684T + 71T^{2} \) |
| 73 | \( 1 - 2T + 73T^{2} \) |
| 79 | \( 1 + 6.24T + 79T^{2} \) |
| 83 | \( 1 - 12.6T + 83T^{2} \) |
| 89 | \( 1 + 5.12T + 89T^{2} \) |
| 97 | \( 1 + 6T + 97T^{2} \) |
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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
−7.56971372418466804539927384322, −6.63240962341754208822362135949, −6.15734314558101067821059669810, −5.31884386453976846140499991123, −4.72628109769820027393430279812, −3.98423927982216671634500167293, −2.95794496620772963152941574193, −2.19342976160115189162269008063, −1.38803074314310923480415015236, 0,
1.38803074314310923480415015236, 2.19342976160115189162269008063, 2.95794496620772963152941574193, 3.98423927982216671634500167293, 4.72628109769820027393430279812, 5.31884386453976846140499991123, 6.15734314558101067821059669810, 6.63240962341754208822362135949, 7.56971372418466804539927384322
