| L(s) = 1 | + 2-s + 4-s + 2·7-s + 8-s − 4·13-s + 2·14-s + 16-s + 6·17-s + 4·19-s + 6·23-s − 4·26-s + 2·28-s + 6·29-s + 8·31-s + 32-s + 6·34-s + 10·37-s + 4·38-s + 6·41-s + 8·43-s + 6·46-s − 6·47-s − 3·49-s − 4·52-s + 2·56-s + 6·58-s − 8·61-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s + 0.755·7-s + 0.353·8-s − 1.10·13-s + 0.534·14-s + 1/4·16-s + 1.45·17-s + 0.917·19-s + 1.25·23-s − 0.784·26-s + 0.377·28-s + 1.11·29-s + 1.43·31-s + 0.176·32-s + 1.02·34-s + 1.64·37-s + 0.648·38-s + 0.937·41-s + 1.21·43-s + 0.884·46-s − 0.875·47-s − 3/7·49-s − 0.554·52-s + 0.267·56-s + 0.787·58-s − 1.02·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 54450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 54450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.957134636\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.957134636\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| 11 | \( 1 \) |
| good | 7 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 8 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + 6 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 + 14 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 + 14 T + p T^{2} \) |
| 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
−14.37514846726047, −14.10275168100768, −13.46409878900677, −12.83366977895796, −12.38084945797546, −11.94998983113647, −11.45648809665936, −11.02709669956321, −10.31190512827768, −9.812470020589678, −9.436543109325477, −8.581637679820715, −7.971850161973466, −7.512507897144904, −7.208932209883659, −6.268553513575817, −5.902373607295450, −5.081091218821238, −4.797553525165767, −4.330281889459990, −3.336976517692873, −2.868728345793669, −2.355707318640834, −1.244748463243351, −0.8679066291062329,
0.8679066291062329, 1.244748463243351, 2.355707318640834, 2.868728345793669, 3.336976517692873, 4.330281889459990, 4.797553525165767, 5.081091218821238, 5.902373607295450, 6.268553513575817, 7.208932209883659, 7.512507897144904, 7.971850161973466, 8.581637679820715, 9.436543109325477, 9.812470020589678, 10.31190512827768, 11.02709669956321, 11.45648809665936, 11.94998983113647, 12.38084945797546, 12.83366977895796, 13.46409878900677, 14.10275168100768, 14.37514846726047
