| L(s) = 1 | + 3-s − 5-s − 7-s + 9-s + 11-s + 2·13-s − 15-s − 2·17-s − 21-s + 4·23-s + 25-s + 27-s + 6·29-s − 4·31-s + 33-s + 35-s − 10·37-s + 2·39-s − 2·41-s + 4·43-s − 45-s − 8·47-s + 49-s − 2·51-s + 2·53-s − 55-s − 4·59-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.447·5-s − 0.377·7-s + 1/3·9-s + 0.301·11-s + 0.554·13-s − 0.258·15-s − 0.485·17-s − 0.218·21-s + 0.834·23-s + 1/5·25-s + 0.192·27-s + 1.11·29-s − 0.718·31-s + 0.174·33-s + 0.169·35-s − 1.64·37-s + 0.320·39-s − 0.312·41-s + 0.609·43-s − 0.149·45-s − 1.16·47-s + 1/7·49-s − 0.280·51-s + 0.274·53-s − 0.134·55-s − 0.520·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 18480 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 18480 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 - T \) |
| good | 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 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
−15.76624265020798, −15.64276169326409, −14.98204854984057, −14.37881363810621, −13.88921172275037, −13.29461677423475, −12.84256617750107, −12.18119447005485, −11.71652524897387, −10.91651205969488, −10.56616282340046, −9.824704019641143, −9.165359265208335, −8.671313488855114, −8.299430269703721, −7.368088486893562, −7.005749378437267, −6.341031550338341, −5.627756100229152, −4.713859855234310, −4.251386859219899, −3.296551205721657, −3.091435697884406, −1.974193669976861, −1.188904827617700, 0,
1.188904827617700, 1.974193669976861, 3.091435697884406, 3.296551205721657, 4.251386859219899, 4.713859855234310, 5.627756100229152, 6.341031550338341, 7.005749378437267, 7.368088486893562, 8.299430269703721, 8.671313488855114, 9.165359265208335, 9.824704019641143, 10.56616282340046, 10.91651205969488, 11.71652524897387, 12.18119447005485, 12.84256617750107, 13.29461677423475, 13.88921172275037, 14.37881363810621, 14.98204854984057, 15.64276169326409, 15.76624265020798
