| L(s) = 1 | − 2-s − 1.82·3-s + 4-s − 3.41·5-s + 1.82·6-s − 0.804·7-s − 8-s + 0.329·9-s + 3.41·10-s + 6.38·11-s − 1.82·12-s + 6.97·13-s + 0.804·14-s + 6.23·15-s + 16-s − 0.0923·17-s − 0.329·18-s + 0.778·19-s − 3.41·20-s + 1.46·21-s − 6.38·22-s − 9.02·23-s + 1.82·24-s + 6.66·25-s − 6.97·26-s + 4.87·27-s − 0.804·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.05·3-s + 0.5·4-s − 1.52·5-s + 0.744·6-s − 0.303·7-s − 0.353·8-s + 0.109·9-s + 1.07·10-s + 1.92·11-s − 0.526·12-s + 1.93·13-s + 0.214·14-s + 1.60·15-s + 0.250·16-s − 0.0224·17-s − 0.0776·18-s + 0.178·19-s − 0.763·20-s + 0.320·21-s − 1.36·22-s − 1.88·23-s + 0.372·24-s + 1.33·25-s − 1.36·26-s + 0.937·27-s − 0.151·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4034 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4034 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6763976631\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6763976631\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 2017 | \( 1 - T \) |
| good | 3 | \( 1 + 1.82T + 3T^{2} \) |
| 5 | \( 1 + 3.41T + 5T^{2} \) |
| 7 | \( 1 + 0.804T + 7T^{2} \) |
| 11 | \( 1 - 6.38T + 11T^{2} \) |
| 13 | \( 1 - 6.97T + 13T^{2} \) |
| 17 | \( 1 + 0.0923T + 17T^{2} \) |
| 19 | \( 1 - 0.778T + 19T^{2} \) |
| 23 | \( 1 + 9.02T + 23T^{2} \) |
| 29 | \( 1 - 6.55T + 29T^{2} \) |
| 31 | \( 1 - 7.08T + 31T^{2} \) |
| 37 | \( 1 - 0.920T + 37T^{2} \) |
| 41 | \( 1 + 5.27T + 41T^{2} \) |
| 43 | \( 1 - 5.97T + 43T^{2} \) |
| 47 | \( 1 - 0.703T + 47T^{2} \) |
| 53 | \( 1 + 2.84T + 53T^{2} \) |
| 59 | \( 1 - 0.732T + 59T^{2} \) |
| 61 | \( 1 - 9.38T + 61T^{2} \) |
| 67 | \( 1 + 1.11T + 67T^{2} \) |
| 71 | \( 1 + 1.33T + 71T^{2} \) |
| 73 | \( 1 + 14.3T + 73T^{2} \) |
| 79 | \( 1 - 1.37T + 79T^{2} \) |
| 83 | \( 1 + 12.4T + 83T^{2} \) |
| 89 | \( 1 + 0.0384T + 89T^{2} \) |
| 97 | \( 1 - 14.8T + 97T^{2} \) |
| 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
−8.473054343121847522207822794955, −7.87512173307445447785147544856, −6.75913299207954080306739737728, −6.39986370473840262266020375093, −5.85214864205146104106548752301, −4.43118376558513863886313009993, −3.92699294842833705523682263063, −3.15727978931742646894744699633, −1.39022226947423721893146205848, −0.62065828455608655723128938004,
0.62065828455608655723128938004, 1.39022226947423721893146205848, 3.15727978931742646894744699633, 3.92699294842833705523682263063, 4.43118376558513863886313009993, 5.85214864205146104106548752301, 6.39986370473840262266020375093, 6.75913299207954080306739737728, 7.87512173307445447785147544856, 8.473054343121847522207822794955
