| L(s) = 1 | − 8·3-s + 10·5-s + 49·7-s − 179·9-s + 340·11-s − 294·13-s − 80·15-s + 1.22e3·17-s − 2.43e3·19-s − 392·21-s − 2.00e3·23-s − 3.02e3·25-s + 3.37e3·27-s − 6.74e3·29-s − 8.85e3·31-s − 2.72e3·33-s + 490·35-s + 9.18e3·37-s + 2.35e3·39-s − 1.45e4·41-s − 8.10e3·43-s − 1.79e3·45-s + 312·47-s + 2.40e3·49-s − 9.80e3·51-s − 1.46e4·53-s + 3.40e3·55-s + ⋯ |
| L(s) = 1 | − 0.513·3-s + 0.178·5-s + 0.377·7-s − 0.736·9-s + 0.847·11-s − 0.482·13-s − 0.0918·15-s + 1.02·17-s − 1.54·19-s − 0.193·21-s − 0.788·23-s − 0.967·25-s + 0.891·27-s − 1.48·29-s − 1.65·31-s − 0.434·33-s + 0.0676·35-s + 1.10·37-s + 0.247·39-s − 1.35·41-s − 0.668·43-s − 0.131·45-s + 0.0206·47-s + 1/7·49-s − 0.528·51-s − 0.715·53-s + 0.151·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 112 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 112 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 - p^{2} T \) |
| good | 3 | \( 1 + 8 T + p^{5} T^{2} \) |
| 5 | \( 1 - 2 p T + p^{5} T^{2} \) |
| 11 | \( 1 - 340 T + p^{5} T^{2} \) |
| 13 | \( 1 + 294 T + p^{5} T^{2} \) |
| 17 | \( 1 - 1226 T + p^{5} T^{2} \) |
| 19 | \( 1 + 128 p T + p^{5} T^{2} \) |
| 23 | \( 1 + 2000 T + p^{5} T^{2} \) |
| 29 | \( 1 + 6746 T + p^{5} T^{2} \) |
| 31 | \( 1 + 8856 T + p^{5} T^{2} \) |
| 37 | \( 1 - 9182 T + p^{5} T^{2} \) |
| 41 | \( 1 + 14574 T + p^{5} T^{2} \) |
| 43 | \( 1 + 8108 T + p^{5} T^{2} \) |
| 47 | \( 1 - 312 T + p^{5} T^{2} \) |
| 53 | \( 1 + 14634 T + p^{5} T^{2} \) |
| 59 | \( 1 - 27656 T + p^{5} T^{2} \) |
| 61 | \( 1 - 34338 T + p^{5} T^{2} \) |
| 67 | \( 1 + 12316 T + p^{5} T^{2} \) |
| 71 | \( 1 + 520 p T + p^{5} T^{2} \) |
| 73 | \( 1 + 61718 T + p^{5} T^{2} \) |
| 79 | \( 1 - 64752 T + p^{5} T^{2} \) |
| 83 | \( 1 - 77056 T + p^{5} T^{2} \) |
| 89 | \( 1 + 8166 T + p^{5} T^{2} \) |
| 97 | \( 1 - 20650 T + p^{5} T^{2} \) |
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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
−11.97969046734305095771329069886, −11.28062968373289447958654151256, −10.11018779002133801597096172979, −8.936973613449736599006259883745, −7.74405471678358397992031024770, −6.31314561263596488999738783962, −5.35492972362475407178243831365, −3.82227743296797929142567636949, −1.91170664168361577824095044432, 0,
1.91170664168361577824095044432, 3.82227743296797929142567636949, 5.35492972362475407178243831365, 6.31314561263596488999738783962, 7.74405471678358397992031024770, 8.936973613449736599006259883745, 10.11018779002133801597096172979, 11.28062968373289447958654151256, 11.97969046734305095771329069886
