L(s) = 1 | + 2·5-s + 11-s + 2·13-s − 3·17-s − 7·19-s − 7·23-s − 25-s − 5·29-s + 2·31-s − 3·37-s + 6·41-s + 11·43-s + 7·47-s − 4·53-s + 2·55-s + 11·59-s + 10·61-s + 4·65-s − 4·67-s − 5·71-s + 8·73-s + 8·79-s + 14·83-s − 6·85-s + 2·89-s − 14·95-s − 15·97-s + ⋯ |
L(s) = 1 | + 0.894·5-s + 0.301·11-s + 0.554·13-s − 0.727·17-s − 1.60·19-s − 1.45·23-s − 1/5·25-s − 0.928·29-s + 0.359·31-s − 0.493·37-s + 0.937·41-s + 1.67·43-s + 1.02·47-s − 0.549·53-s + 0.269·55-s + 1.43·59-s + 1.28·61-s + 0.496·65-s − 0.488·67-s − 0.593·71-s + 0.936·73-s + 0.900·79-s + 1.53·83-s − 0.650·85-s + 0.211·89-s − 1.43·95-s − 1.52·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 310464 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 310464 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.222686392\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.222686392\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
| 11 | \( 1 - T \) |
good | 5 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 19 | \( 1 + 7 T + p T^{2} \) |
| 23 | \( 1 + 7 T + p T^{2} \) |
| 29 | \( 1 + 5 T + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 + 3 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 11 T + p T^{2} \) |
| 47 | \( 1 - 7 T + p T^{2} \) |
| 53 | \( 1 + 4 T + p T^{2} \) |
| 59 | \( 1 - 11 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + 5 T + p T^{2} \) |
| 73 | \( 1 - 8 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 - 14 T + p T^{2} \) |
| 89 | \( 1 - 2 T + p T^{2} \) |
| 97 | \( 1 + 15 T + p T^{2} \) |
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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
−12.70141659234930, −12.24170963351099, −11.82096758424169, −11.07950682582792, −10.86136685004897, −10.45381584518413, −9.851133399654097, −9.452066779967904, −9.020057866911902, −8.599970105363708, −8.034263142182938, −7.665297145116965, −6.878481504476083, −6.492282495368364, −6.127008263884077, −5.617958346717125, −5.281814343008364, −4.274658932285395, −4.127937889938476, −3.700308542519119, −2.700486387716396, −2.180923597992556, −1.987451808861945, −1.177072933636803, −0.3913450652745051,
0.3913450652745051, 1.177072933636803, 1.987451808861945, 2.180923597992556, 2.700486387716396, 3.700308542519119, 4.127937889938476, 4.274658932285395, 5.281814343008364, 5.617958346717125, 6.127008263884077, 6.492282495368364, 6.878481504476083, 7.665297145116965, 8.034263142182938, 8.599970105363708, 9.020057866911902, 9.452066779967904, 9.851133399654097, 10.45381584518413, 10.86136685004897, 11.07950682582792, 11.82096758424169, 12.24170963351099, 12.70141659234930