| L(s) = 1 | − 2-s + 0.772·3-s + 4-s − 1.40·5-s − 0.772·6-s + 7-s − 8-s − 2.40·9-s + 1.40·10-s − 11-s + 0.772·12-s − 13-s − 14-s − 1.08·15-s + 16-s + 0.772·17-s + 2.40·18-s + 3.40·19-s − 1.40·20-s + 0.772·21-s + 22-s + 2.45·23-s − 0.772·24-s − 3.03·25-s + 26-s − 4.17·27-s + 28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.446·3-s + 0.5·4-s − 0.627·5-s − 0.315·6-s + 0.377·7-s − 0.353·8-s − 0.800·9-s + 0.443·10-s − 0.301·11-s + 0.223·12-s − 0.277·13-s − 0.267·14-s − 0.279·15-s + 0.250·16-s + 0.187·17-s + 0.566·18-s + 0.780·19-s − 0.313·20-s + 0.168·21-s + 0.213·22-s + 0.511·23-s − 0.157·24-s − 0.606·25-s + 0.196·26-s − 0.803·27-s + 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2002 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2002 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.150553308\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.150553308\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 + T \) |
| 13 | \( 1 + T \) |
| good | 3 | \( 1 - 0.772T + 3T^{2} \) |
| 5 | \( 1 + 1.40T + 5T^{2} \) |
| 17 | \( 1 - 0.772T + 17T^{2} \) |
| 19 | \( 1 - 3.40T + 19T^{2} \) |
| 23 | \( 1 - 2.45T + 23T^{2} \) |
| 29 | \( 1 - 6.80T + 29T^{2} \) |
| 31 | \( 1 + 4.35T + 31T^{2} \) |
| 37 | \( 1 + 37T^{2} \) |
| 41 | \( 1 - 10.3T + 41T^{2} \) |
| 43 | \( 1 - 2.31T + 43T^{2} \) |
| 47 | \( 1 - 9.54T + 47T^{2} \) |
| 53 | \( 1 - 8.03T + 53T^{2} \) |
| 59 | \( 1 - 3.54T + 59T^{2} \) |
| 61 | \( 1 + 10.4T + 61T^{2} \) |
| 67 | \( 1 + 0.772T + 67T^{2} \) |
| 71 | \( 1 - 13.7T + 71T^{2} \) |
| 73 | \( 1 - 7.25T + 73T^{2} \) |
| 79 | \( 1 + 9.75T + 79T^{2} \) |
| 83 | \( 1 - 0.311T + 83T^{2} \) |
| 89 | \( 1 - 4.77T + 89T^{2} \) |
| 97 | \( 1 - 5.25T + 97T^{2} \) |
| 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
−9.047564430338949017297358477946, −8.379190924397781320646425492560, −7.69521188511165692525287935398, −7.22961061098375477519075868031, −6.01065467575138829184909665929, −5.23801436315775963475139161279, −4.08360374061192247226653583760, −3.07896732007676642341811263921, −2.26858977004330598215521926252, −0.76539516206949180344001996983,
0.76539516206949180344001996983, 2.26858977004330598215521926252, 3.07896732007676642341811263921, 4.08360374061192247226653583760, 5.23801436315775963475139161279, 6.01065467575138829184909665929, 7.22961061098375477519075868031, 7.69521188511165692525287935398, 8.379190924397781320646425492560, 9.047564430338949017297358477946
