| L(s) = 1 | + 2-s + 4-s + 7-s + 8-s − 3·9-s + 6·13-s + 14-s + 16-s + 4·17-s − 3·18-s + 4·19-s − 4·23-s + 6·26-s + 28-s − 2·31-s + 32-s + 4·34-s − 3·36-s − 2·37-s + 4·38-s − 10·41-s − 8·43-s − 4·46-s + 49-s + 6·52-s − 2·53-s + 56-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s + 0.377·7-s + 0.353·8-s − 9-s + 1.66·13-s + 0.267·14-s + 1/4·16-s + 0.970·17-s − 0.707·18-s + 0.917·19-s − 0.834·23-s + 1.17·26-s + 0.188·28-s − 0.359·31-s + 0.176·32-s + 0.685·34-s − 1/2·36-s − 0.328·37-s + 0.648·38-s − 1.56·41-s − 1.21·43-s − 0.589·46-s + 1/7·49-s + 0.832·52-s − 0.274·53-s + 0.133·56-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 42350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 42350 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 5 | \( 1 \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 \) |
| good | 3 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 6 T + p T^{2} \) |
| 17 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 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 | \( 1 + 2 T + p T^{2} \) |
| 71 | \( 1 + 4 T + p T^{2} \) |
| 73 | \( 1 + 4 T + p T^{2} \) |
| 79 | \( 1 + 14 T + p T^{2} \) |
| 83 | \( 1 + 18 T + p T^{2} \) |
| 89 | \( 1 - 14 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
| 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
−14.81965966576570, −14.34716950368111, −13.97581267817399, −13.47037879082200, −13.13480627340809, −12.21460671445841, −11.88493196395414, −11.45796375876639, −10.99024181602547, −10.31525901158405, −9.925507247716968, −8.962300071670260, −8.630439139242772, −7.991035392657952, −7.575333808125760, −6.756394838531736, −6.138683696366280, −5.709542639489279, −5.251297042256123, −4.566110681113744, −3.696008125470394, −3.358666154402211, −2.785577818776674, −1.684131137758636, −1.288964260887400, 0,
1.288964260887400, 1.684131137758636, 2.785577818776674, 3.358666154402211, 3.696008125470394, 4.566110681113744, 5.251297042256123, 5.709542639489279, 6.138683696366280, 6.756394838531736, 7.575333808125760, 7.991035392657952, 8.630439139242772, 8.962300071670260, 9.925507247716968, 10.31525901158405, 10.99024181602547, 11.45796375876639, 11.88493196395414, 12.21460671445841, 13.13480627340809, 13.47037879082200, 13.97581267817399, 14.34716950368111, 14.81965966576570
