| L(s) = 1 | − 1.80·3-s + 4-s − 1.24·5-s + 2.24·9-s − 11-s − 1.80·12-s + 2.24·15-s + 16-s − 1.24·20-s − 0.445·23-s + 0.554·25-s − 2.24·27-s + 0.445·31-s + 1.80·33-s + 2.24·36-s + 1.80·37-s − 44-s − 2.80·45-s + 0.445·47-s − 1.80·48-s + 49-s + 1.24·53-s + 1.24·55-s + 1.80·59-s + 2.24·60-s + 64-s − 2·67-s + ⋯ |
| L(s) = 1 | − 1.80·3-s + 4-s − 1.24·5-s + 2.24·9-s − 11-s − 1.80·12-s + 2.24·15-s + 16-s − 1.24·20-s − 0.445·23-s + 0.554·25-s − 2.24·27-s + 0.445·31-s + 1.80·33-s + 2.24·36-s + 1.80·37-s − 44-s − 2.80·45-s + 0.445·47-s − 1.80·48-s + 49-s + 1.24·53-s + 1.24·55-s + 1.80·59-s + 2.24·60-s + 64-s − 2·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1859 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1859 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.5743822708\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5743822708\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 11 | \( 1 + T \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 - T^{2} \) |
| 3 | \( 1 + 1.80T + T^{2} \) |
| 5 | \( 1 + 1.24T + T^{2} \) |
| 7 | \( 1 - T^{2} \) |
| 17 | \( 1 - T^{2} \) |
| 19 | \( 1 - T^{2} \) |
| 23 | \( 1 + 0.445T + T^{2} \) |
| 29 | \( 1 - T^{2} \) |
| 31 | \( 1 - 0.445T + T^{2} \) |
| 37 | \( 1 - 1.80T + T^{2} \) |
| 41 | \( 1 - T^{2} \) |
| 43 | \( 1 - T^{2} \) |
| 47 | \( 1 - 0.445T + T^{2} \) |
| 53 | \( 1 - 1.24T + T^{2} \) |
| 59 | \( 1 - 1.80T + T^{2} \) |
| 61 | \( 1 - T^{2} \) |
| 67 | \( 1 + 2T + T^{2} \) |
| 71 | \( 1 - 1.80T + T^{2} \) |
| 73 | \( 1 - T^{2} \) |
| 79 | \( 1 - T^{2} \) |
| 83 | \( 1 - T^{2} \) |
| 89 | \( 1 - 0.445T + T^{2} \) |
| 97 | \( 1 + 1.24T + T^{2} \) |
| 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
−9.868243547823741066176786762245, −8.322025579780836830662388671595, −7.53782990592625448493195984670, −7.10073247821734180567273077415, −6.15202380233172224349046371869, −5.57609618456610737104685267280, −4.63496177075956614333509709364, −3.78961248791494992698185424245, −2.42820827478680535164772596065, −0.817084319002828612197594699503,
0.817084319002828612197594699503, 2.42820827478680535164772596065, 3.78961248791494992698185424245, 4.63496177075956614333509709364, 5.57609618456610737104685267280, 6.15202380233172224349046371869, 7.10073247821734180567273077415, 7.53782990592625448493195984670, 8.322025579780836830662388671595, 9.868243547823741066176786762245
