| L(s) = 1 | + 3-s − 5-s − 2·7-s + 9-s − 0.828·11-s + 0.828·13-s − 15-s + 2.82·17-s − 2·21-s − 5.65·23-s + 25-s + 27-s + 3.65·29-s − 1.17·31-s − 0.828·33-s + 2·35-s − 6.48·37-s + 0.828·39-s − 3.65·41-s + 1.65·43-s − 45-s + 1.65·47-s − 3·49-s + 2.82·51-s − 11.6·53-s + 0.828·55-s + 4.82·59-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.447·5-s − 0.755·7-s + 0.333·9-s − 0.249·11-s + 0.229·13-s − 0.258·15-s + 0.685·17-s − 0.436·21-s − 1.17·23-s + 0.200·25-s + 0.192·27-s + 0.679·29-s − 0.210·31-s − 0.144·33-s + 0.338·35-s − 1.06·37-s + 0.132·39-s − 0.571·41-s + 0.252·43-s − 0.149·45-s + 0.241·47-s − 0.428·49-s + 0.396·51-s − 1.60·53-s + 0.111·55-s + 0.628·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3840 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| good | 7 | \( 1 + 2T + 7T^{2} \) |
| 11 | \( 1 + 0.828T + 11T^{2} \) |
| 13 | \( 1 - 0.828T + 13T^{2} \) |
| 17 | \( 1 - 2.82T + 17T^{2} \) |
| 19 | \( 1 + 19T^{2} \) |
| 23 | \( 1 + 5.65T + 23T^{2} \) |
| 29 | \( 1 - 3.65T + 29T^{2} \) |
| 31 | \( 1 + 1.17T + 31T^{2} \) |
| 37 | \( 1 + 6.48T + 37T^{2} \) |
| 41 | \( 1 + 3.65T + 41T^{2} \) |
| 43 | \( 1 - 1.65T + 43T^{2} \) |
| 47 | \( 1 - 1.65T + 47T^{2} \) |
| 53 | \( 1 + 11.6T + 53T^{2} \) |
| 59 | \( 1 - 4.82T + 59T^{2} \) |
| 61 | \( 1 - 9.65T + 61T^{2} \) |
| 67 | \( 1 + 9.65T + 67T^{2} \) |
| 71 | \( 1 + 13.6T + 71T^{2} \) |
| 73 | \( 1 - 9.31T + 73T^{2} \) |
| 79 | \( 1 + 12.4T + 79T^{2} \) |
| 83 | \( 1 + 83T^{2} \) |
| 89 | \( 1 + 4.34T + 89T^{2} \) |
| 97 | \( 1 - 7.65T + 97T^{2} \) |
| 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
−8.162244655544744862092529867049, −7.48206244061132668655557186412, −6.72299191914191127814883950879, −5.97784527345172530967753789503, −5.06421808955392790753205419238, −4.07509436663480773299586041954, −3.41947163587125376660483214850, −2.67529815770567707012049907024, −1.48670746216841464543140866380, 0,
1.48670746216841464543140866380, 2.67529815770567707012049907024, 3.41947163587125376660483214850, 4.07509436663480773299586041954, 5.06421808955392790753205419238, 5.97784527345172530967753789503, 6.72299191914191127814883950879, 7.48206244061132668655557186412, 8.162244655544744862092529867049
