| L(s) = 1 | − 4-s + 16-s − 4·19-s − 12·29-s + 16·31-s + 12·41-s − 49-s − 24·59-s + 16·61-s − 64-s + 12·71-s + 4·76-s + 32·79-s − 12·89-s + 12·101-s + 20·109-s + 12·116-s − 22·121-s − 16·124-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + ⋯ |
| L(s) = 1 | − 1/2·4-s + 1/4·16-s − 0.917·19-s − 2.22·29-s + 2.87·31-s + 1.87·41-s − 1/7·49-s − 3.12·59-s + 2.04·61-s − 1/8·64-s + 1.42·71-s + 0.458·76-s + 3.60·79-s − 1.27·89-s + 1.19·101-s + 1.91·109-s + 1.11·116-s − 2·121-s − 1.43·124-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9922500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9922500 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.778206092\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.778206092\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | $C_2$ | \( 1 + T^{2} \) |
| 3 | | \( 1 \) |
| 5 | | \( 1 \) |
| 7 | $C_2$ | \( 1 + T^{2} \) |
| good | 11 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 58 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 82 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 58 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 130 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 142 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 16 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 94 T^{2} + p^{2} T^{4} \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.913588742562795045312389790636, −8.489250078825845077535406397302, −7.969147393638267170447910365320, −7.915321324046354594827067599451, −7.53524729916976273244206017927, −6.96553250866301347378844342410, −6.53678653285585313234974678592, −6.16171876502182176338580026076, −6.01343750212412150270780751421, −5.33040325595765643015636409375, −5.06586964490221257497733588172, −4.44042058937328080571874913564, −4.39717522695574308172824452433, −3.59135396476997925927508395011, −3.58830600477777672384790510525, −2.68763404797971428976602987156, −2.40296417458936995675272446746, −1.82744132002226770271992457725, −1.07208474722688295718326417191, −0.47436319440764993860101528223,
0.47436319440764993860101528223, 1.07208474722688295718326417191, 1.82744132002226770271992457725, 2.40296417458936995675272446746, 2.68763404797971428976602987156, 3.58830600477777672384790510525, 3.59135396476997925927508395011, 4.39717522695574308172824452433, 4.44042058937328080571874913564, 5.06586964490221257497733588172, 5.33040325595765643015636409375, 6.01343750212412150270780751421, 6.16171876502182176338580026076, 6.53678653285585313234974678592, 6.96553250866301347378844342410, 7.53524729916976273244206017927, 7.915321324046354594827067599451, 7.969147393638267170447910365320, 8.489250078825845077535406397302, 8.913588742562795045312389790636
