| L(s) = 1 | − 2-s − 4-s + 5-s + 3·8-s − 3·9-s − 10-s + 4·11-s − 2·13-s − 16-s − 6·17-s + 3·18-s − 20-s − 4·22-s + 4·23-s + 25-s + 2·26-s − 2·29-s + 4·31-s − 5·32-s + 6·34-s + 3·36-s − 10·37-s + 3·40-s − 6·41-s + 4·43-s − 4·44-s − 3·45-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1/2·4-s + 0.447·5-s + 1.06·8-s − 9-s − 0.316·10-s + 1.20·11-s − 0.554·13-s − 1/4·16-s − 1.45·17-s + 0.707·18-s − 0.223·20-s − 0.852·22-s + 0.834·23-s + 1/5·25-s + 0.392·26-s − 0.371·29-s + 0.718·31-s − 0.883·32-s + 1.02·34-s + 1/2·36-s − 1.64·37-s + 0.474·40-s − 0.937·41-s + 0.609·43-s − 0.603·44-s − 0.447·45-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2005 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2005 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 - T \) |
| 401 | \( 1 - T \) |
| good | 2 | \( 1 + T + p T^{2} \) |
| 3 | \( 1 + p T^{2} \) |
| 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 8 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 + p T^{2} \) |
| 71 | \( 1 - 4 T + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 + 12 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 - 10 T + p T^{2} \) |
| 97 | \( 1 + 6 T + p T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.910159619650622296165115496591, −8.344221776937260259402405455960, −7.18606324297450196513347237566, −6.58387442700360860775917772549, −5.53061415107159207269254211386, −4.73263370021136221235882375157, −3.83064612679619982339324173145, −2.57947891595731839507631448040, −1.42475002819210777825730774545, 0,
1.42475002819210777825730774545, 2.57947891595731839507631448040, 3.83064612679619982339324173145, 4.73263370021136221235882375157, 5.53061415107159207269254211386, 6.58387442700360860775917772549, 7.18606324297450196513347237566, 8.344221776937260259402405455960, 8.910159619650622296165115496591
