| L(s) = 1 | + 3-s + 5-s + 7-s + 9-s − 4·11-s − 2·13-s + 15-s + 2·17-s − 4·19-s + 21-s − 23-s + 25-s + 27-s − 2·29-s + 8·31-s − 4·33-s + 35-s + 6·37-s − 2·39-s − 6·41-s − 4·43-s + 45-s + 8·47-s + 49-s + 2·51-s − 10·53-s − 4·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.447·5-s + 0.377·7-s + 1/3·9-s − 1.20·11-s − 0.554·13-s + 0.258·15-s + 0.485·17-s − 0.917·19-s + 0.218·21-s − 0.208·23-s + 1/5·25-s + 0.192·27-s − 0.371·29-s + 1.43·31-s − 0.696·33-s + 0.169·35-s + 0.986·37-s − 0.320·39-s − 0.937·41-s − 0.609·43-s + 0.149·45-s + 1.16·47-s + 1/7·49-s + 0.280·51-s − 1.37·53-s − 0.539·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 19320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 19320 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 23 | \( 1 + T \) |
| good | 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 - 14 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 - 16 T + p T^{2} \) |
| 73 | \( 1 + 14 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 + 14 T + p T^{2} \) |
| 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
−15.88582962243242, −15.28861433154526, −14.95185587735242, −14.33575448755656, −13.75250434876020, −13.38854999145867, −12.66546156213351, −12.37114149870741, −11.52038537280413, −10.93976434011239, −10.17748868855932, −10.03134311309943, −9.282486716067283, −8.545329000082698, −8.088428359183361, −7.616046270553266, −6.911683528648977, −6.168581531091734, −5.543287624117810, −4.801542132477957, −4.372110345104147, −3.355390404173569, −2.650532265030181, −2.161010215303590, −1.247710192829133, 0,
1.247710192829133, 2.161010215303590, 2.650532265030181, 3.355390404173569, 4.372110345104147, 4.801542132477957, 5.543287624117810, 6.168581531091734, 6.911683528648977, 7.616046270553266, 8.088428359183361, 8.545329000082698, 9.282486716067283, 10.03134311309943, 10.17748868855932, 10.93976434011239, 11.52038537280413, 12.37114149870741, 12.66546156213351, 13.38854999145867, 13.75250434876020, 14.33575448755656, 14.95185587735242, 15.28861433154526, 15.88582962243242
