L(s) = 1 | + 4·11-s − 4·19-s + 12·29-s − 4·31-s + 20·41-s − 49-s − 16·59-s − 4·61-s + 20·71-s + 4·89-s + 12·101-s + 28·109-s − 10·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 10·169-s + 173-s + 179-s + 181-s + 191-s + ⋯ |
L(s) = 1 | + 1.20·11-s − 0.917·19-s + 2.22·29-s − 0.718·31-s + 3.12·41-s − 1/7·49-s − 2.08·59-s − 0.512·61-s + 2.37·71-s + 0.423·89-s + 1.19·101-s + 2.68·109-s − 0.909·121-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 + 0.0773·167-s + 0.769·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 39690000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 39690000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.829230816\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.829230816\) |
\(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 3 | | \( 1 \) |
| 5 | | \( 1 \) |
| 7 | $C_2$ | \( 1 + T^{2} \) |
good | 11 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 30 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 + 26 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 + 58 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 30 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 130 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 18 T + p T^{2} )( 1 + 18 T + p T^{2} ) \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.091172535600355663489655844032, −7.937671080429113715282872728935, −7.39064780084228608815784087944, −7.35056163543842104933475572535, −6.55453481382828141098344411268, −6.46028655286727064212423854705, −6.23077888690775140778736351497, −5.93598055542009928829687955892, −5.17571917677945202904205026067, −5.11542120668297631948622964551, −4.44478499298663224056762754507, −4.20668315968916506154938187359, −4.00647991642999781317710018220, −3.42244125722796490654879978995, −2.81508504838901768470439952619, −2.75739453643418687511031109826, −1.89907005178994095225488619318, −1.74391184899707157466825444562, −0.858282277204015813740215577933, −0.64164850107965483703340264522,
0.64164850107965483703340264522, 0.858282277204015813740215577933, 1.74391184899707157466825444562, 1.89907005178994095225488619318, 2.75739453643418687511031109826, 2.81508504838901768470439952619, 3.42244125722796490654879978995, 4.00647991642999781317710018220, 4.20668315968916506154938187359, 4.44478499298663224056762754507, 5.11542120668297631948622964551, 5.17571917677945202904205026067, 5.93598055542009928829687955892, 6.23077888690775140778736351497, 6.46028655286727064212423854705, 6.55453481382828141098344411268, 7.35056163543842104933475572535, 7.39064780084228608815784087944, 7.937671080429113715282872728935, 8.091172535600355663489655844032