| L(s) = 1 | + 3-s − 2·5-s − 7-s + 9-s + 4·11-s − 2·13-s − 2·15-s − 6·17-s − 4·19-s − 21-s + 23-s − 25-s + 27-s − 2·29-s + 8·31-s + 4·33-s + 2·35-s + 6·37-s − 2·39-s − 6·41-s + 4·43-s − 2·45-s + 8·47-s + 49-s − 6·51-s + 6·53-s − 8·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.894·5-s − 0.377·7-s + 1/3·9-s + 1.20·11-s − 0.554·13-s − 0.516·15-s − 1.45·17-s − 0.917·19-s − 0.218·21-s + 0.208·23-s − 1/5·25-s + 0.192·27-s − 0.371·29-s + 1.43·31-s + 0.696·33-s + 0.338·35-s + 0.986·37-s − 0.320·39-s − 0.937·41-s + 0.609·43-s − 0.298·45-s + 1.16·47-s + 1/7·49-s − 0.840·51-s + 0.824·53-s − 1.07·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7728 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.655316589\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.655316589\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 23 | \( 1 - T \) |
| good | 5 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 - 2 T + p T^{2} \) |
| 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
−7.85576196125267762191993718040, −7.21139972549870270187877373452, −6.57030732616109135001481464416, −6.02500492025265710797266941625, −4.65031396001729753899180856960, −4.28743866446849773696891630830, −3.62971996605385422357186101166, −2.70929230388985105184214574237, −1.92268539533541913400429448327, −0.60792995259228108189590663814,
0.60792995259228108189590663814, 1.92268539533541913400429448327, 2.70929230388985105184214574237, 3.62971996605385422357186101166, 4.28743866446849773696891630830, 4.65031396001729753899180856960, 6.02500492025265710797266941625, 6.57030732616109135001481464416, 7.21139972549870270187877373452, 7.85576196125267762191993718040
