| L(s) = 1 | + 3-s + 9-s − 4·11-s + 2·13-s − 2·17-s + 4·23-s + 27-s + 2·29-s − 8·31-s − 4·33-s − 10·37-s + 2·39-s − 2·41-s − 4·43-s + 4·47-s − 2·51-s − 10·53-s − 4·59-s + 2·61-s − 4·67-s + 4·69-s − 6·73-s + 8·79-s + 81-s − 4·83-s + 2·87-s + 6·89-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1/3·9-s − 1.20·11-s + 0.554·13-s − 0.485·17-s + 0.834·23-s + 0.192·27-s + 0.371·29-s − 1.43·31-s − 0.696·33-s − 1.64·37-s + 0.320·39-s − 0.312·41-s − 0.609·43-s + 0.583·47-s − 0.280·51-s − 1.37·53-s − 0.520·59-s + 0.256·61-s − 0.488·67-s + 0.481·69-s − 0.702·73-s + 0.900·79-s + 1/9·81-s − 0.439·83-s + 0.214·87-s + 0.635·89-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 117600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 117600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.612716576\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.612716576\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 \) |
| 7 | \( 1 \) |
| good | 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 4 T + p T^{2} \) |
| 53 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 - 10 T + p T^{2} \) |
| 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
−13.63181889967855, −13.04548913076318, −12.85389527775203, −12.20268889204407, −11.69246525657539, −10.95337145510346, −10.64855363297432, −10.36534294493629, −9.498209130097124, −9.208607094729867, −8.625556713818920, −8.196172743561388, −7.736934511239595, −7.064477174463314, −6.798443945250820, −6.021358905525550, −5.393643713876576, −5.004476320819926, −4.386037380743543, −3.665615425662381, −3.182153573575374, −2.673650079854607, −1.915051851475133, −1.432278987109087, −0.3586099478636965,
0.3586099478636965, 1.432278987109087, 1.915051851475133, 2.673650079854607, 3.182153573575374, 3.665615425662381, 4.386037380743543, 5.004476320819926, 5.393643713876576, 6.021358905525550, 6.798443945250820, 7.064477174463314, 7.736934511239595, 8.196172743561388, 8.625556713818920, 9.208607094729867, 9.498209130097124, 10.36534294493629, 10.64855363297432, 10.95337145510346, 11.69246525657539, 12.20268889204407, 12.85389527775203, 13.04548913076318, 13.63181889967855
