| L(s) = 1 | + 3-s − 2·4-s − 5-s + 2·7-s + 9-s + 3·11-s − 2·12-s + 2·13-s − 15-s + 4·16-s + 2·19-s + 2·20-s + 2·21-s + 3·23-s + 25-s + 27-s − 4·28-s − 29-s + 8·31-s + 3·33-s − 2·35-s − 2·36-s − 37-s + 2·39-s − 3·41-s − 43-s − 6·44-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 4-s − 0.447·5-s + 0.755·7-s + 1/3·9-s + 0.904·11-s − 0.577·12-s + 0.554·13-s − 0.258·15-s + 16-s + 0.458·19-s + 0.447·20-s + 0.436·21-s + 0.625·23-s + 1/5·25-s + 0.192·27-s − 0.755·28-s − 0.185·29-s + 1.43·31-s + 0.522·33-s − 0.338·35-s − 1/3·36-s − 0.164·37-s + 0.320·39-s − 0.468·41-s − 0.152·43-s − 0.904·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 435 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 435 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.466938990\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.466938990\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 29 | \( 1 + T \) |
| good | 2 | \( 1 + p T^{2} \) |
| 7 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 - 3 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 - 3 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 + T + p T^{2} \) |
| 41 | \( 1 + 3 T + p T^{2} \) |
| 43 | \( 1 + T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 + 3 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 - 8 T + p T^{2} \) |
| 67 | \( 1 - 14 T + p T^{2} \) |
| 71 | \( 1 + 6 T + p T^{2} \) |
| 73 | \( 1 + 7 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 - 9 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 - 11 T + p T^{2} \) |
| 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
−11.21432531928699822637672640237, −10.05944527261858513975292871962, −9.171876011593674243647849968841, −8.450210472115657969983674844783, −7.78896325202640634293492702134, −6.53455962727190409399555438185, −5.10220841214218755215854486710, −4.24183605846174910586180386651, −3.27115066449271458143091293882, −1.29114861944572476697297571680,
1.29114861944572476697297571680, 3.27115066449271458143091293882, 4.24183605846174910586180386651, 5.10220841214218755215854486710, 6.53455962727190409399555438185, 7.78896325202640634293492702134, 8.450210472115657969983674844783, 9.171876011593674243647849968841, 10.05944527261858513975292871962, 11.21432531928699822637672640237
