| L(s) = 1 | + 2-s + 4-s + 5-s + 7-s + 8-s + 10-s + 3·11-s + 5·13-s + 14-s + 16-s − 7·19-s + 20-s + 3·22-s + 6·23-s + 25-s + 5·26-s + 28-s − 6·29-s − 4·31-s + 32-s + 35-s + 2·37-s − 7·38-s + 40-s − 3·41-s − 43-s + 3·44-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s + 0.447·5-s + 0.377·7-s + 0.353·8-s + 0.316·10-s + 0.904·11-s + 1.38·13-s + 0.267·14-s + 1/4·16-s − 1.60·19-s + 0.223·20-s + 0.639·22-s + 1.25·23-s + 1/5·25-s + 0.980·26-s + 0.188·28-s − 1.11·29-s − 0.718·31-s + 0.176·32-s + 0.169·35-s + 0.328·37-s − 1.13·38-s + 0.158·40-s − 0.468·41-s − 0.152·43-s + 0.452·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1890 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1890 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.436561145\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.436561145\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| good | 11 | \( 1 - 3 T + p T^{2} \) |
| 13 | \( 1 - 5 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 + 7 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 3 T + p T^{2} \) |
| 43 | \( 1 + T + p T^{2} \) |
| 47 | \( 1 - 3 T + p T^{2} \) |
| 53 | \( 1 - 9 T + p T^{2} \) |
| 59 | \( 1 - 6 T + p T^{2} \) |
| 61 | \( 1 - 8 T + p T^{2} \) |
| 67 | \( 1 + 13 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 7 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 - 3 T + p T^{2} \) |
| 89 | \( 1 - 9 T + p T^{2} \) |
| 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
−8.926529589016283807355632196941, −8.701856610059850526638171029268, −7.46679210213785799425922090716, −6.62383687933844553298964816100, −6.03267537061710076703648081532, −5.20861398617287979134639009542, −4.17565903266185298934746247405, −3.55919323194744322765656270789, −2.24867210192228587557451291464, −1.28514159815933558570968670699,
1.28514159815933558570968670699, 2.24867210192228587557451291464, 3.55919323194744322765656270789, 4.17565903266185298934746247405, 5.20861398617287979134639009542, 6.03267537061710076703648081532, 6.62383687933844553298964816100, 7.46679210213785799425922090716, 8.701856610059850526638171029268, 8.926529589016283807355632196941
