| L(s) = 1 | + 2·9-s + 6·11-s − 4·16-s + 4·19-s + 12·29-s − 2·31-s + 10·41-s + 14·49-s + 24·59-s + 4·61-s + 4·71-s + 16·79-s − 5·81-s + 8·89-s + 12·99-s − 18·101-s − 14·109-s + 5·121-s + 127-s + 131-s + 137-s + 139-s − 8·144-s + 149-s + 151-s + 157-s + 163-s + ⋯ |
| L(s) = 1 | + 2/3·9-s + 1.80·11-s − 16-s + 0.917·19-s + 2.22·29-s − 0.359·31-s + 1.56·41-s + 2·49-s + 3.12·59-s + 0.512·61-s + 0.474·71-s + 1.80·79-s − 5/9·81-s + 0.847·89-s + 1.20·99-s − 1.79·101-s − 1.34·109-s + 5/11·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 2/3·144-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1155625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1155625 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.973226088\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.973226088\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 5 | | \( 1 \) |
| 43 | $C_2$ | \( 1 + T^{2} \) |
| good | 2 | $C_2$ | \( ( 1 - p T + p T^{2} )( 1 + p T + p T^{2} ) \) |
| 3 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 25 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 45 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 78 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 81 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 125 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 142 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 + 59 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 145 T^{2} + p^{2} T^{4} \) |
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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
−9.887892592088091902631424159776, −9.710887369035426443261044105689, −9.287604040986603424513083812796, −8.885487520408565285330406713451, −8.546681828173521378243024331044, −8.129965403794831182849467838140, −7.29589947142409344562829979625, −7.29369786750731142335093871617, −6.70099659312056784552669654279, −6.41295841238528194521222493379, −5.99468633262745676277317437391, −5.21773867052717109715799487418, −4.97679095399392265340890894007, −4.23583843722116955224224300924, −3.93179156198282917234609400869, −3.63854218768032536755905089511, −2.52108619379382811414155029462, −2.42428671521233822914668873517, −1.23062939970639946144566310459, −0.953298339373886933243585073199,
0.953298339373886933243585073199, 1.23062939970639946144566310459, 2.42428671521233822914668873517, 2.52108619379382811414155029462, 3.63854218768032536755905089511, 3.93179156198282917234609400869, 4.23583843722116955224224300924, 4.97679095399392265340890894007, 5.21773867052717109715799487418, 5.99468633262745676277317437391, 6.41295841238528194521222493379, 6.70099659312056784552669654279, 7.29369786750731142335093871617, 7.29589947142409344562829979625, 8.129965403794831182849467838140, 8.546681828173521378243024331044, 8.885487520408565285330406713451, 9.287604040986603424513083812796, 9.710887369035426443261044105689, 9.887892592088091902631424159776
