| L(s) = 1 | + 2-s + 3-s − 4-s − 5-s + 6-s − 7-s − 3·8-s + 9-s − 10-s + 11-s − 12-s − 2·13-s − 14-s − 15-s − 16-s + 18-s + 4·19-s + 20-s − 21-s + 22-s + 4·23-s − 3·24-s + 25-s − 2·26-s + 27-s + 28-s + 2·29-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s − 1/2·4-s − 0.447·5-s + 0.408·6-s − 0.377·7-s − 1.06·8-s + 1/3·9-s − 0.316·10-s + 0.301·11-s − 0.288·12-s − 0.554·13-s − 0.267·14-s − 0.258·15-s − 1/4·16-s + 0.235·18-s + 0.917·19-s + 0.223·20-s − 0.218·21-s + 0.213·22-s + 0.834·23-s − 0.612·24-s + 1/5·25-s − 0.392·26-s + 0.192·27-s + 0.188·28-s + 0.371·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 333795 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 333795 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 - T \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 - T \) |
| 17 | \( 1 \) |
| good | 2 | \( 1 - T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 + 12 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 14 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 - 16 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 + 16 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 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
−12.96005051868899, −12.49820466313187, −11.98985769772915, −11.51271957043381, −11.33419849392757, −10.35681411975766, −9.973390659618543, −9.688412956774019, −9.128105346977650, −8.649432537233204, −8.365186101844795, −7.745646809285646, −7.213954855385149, −6.811121748405934, −6.273476508965706, −5.708046016370405, −5.053894348415507, −4.823576025943242, −4.219098946444265, −3.738801558634217, −3.204145877214366, −2.901037800165679, −2.309324620548089, −1.364631799848965, −0.7840646492285609, 0,
0.7840646492285609, 1.364631799848965, 2.309324620548089, 2.901037800165679, 3.204145877214366, 3.738801558634217, 4.219098946444265, 4.823576025943242, 5.053894348415507, 5.708046016370405, 6.273476508965706, 6.811121748405934, 7.213954855385149, 7.745646809285646, 8.365186101844795, 8.649432537233204, 9.128105346977650, 9.688412956774019, 9.973390659618543, 10.35681411975766, 11.33419849392757, 11.51271957043381, 11.98985769772915, 12.49820466313187, 12.96005051868899
