| L(s) = 1 | + 4·4-s + 2·9-s + 6·11-s + 12·16-s − 2·19-s − 12·29-s − 8·31-s + 8·36-s − 12·41-s + 24·44-s + 13·49-s + 12·59-s − 2·61-s + 32·64-s + 12·71-s − 8·76-s − 16·79-s − 5·81-s − 24·89-s + 12·99-s + 12·101-s + 32·109-s − 48·116-s + 5·121-s − 32·124-s + 127-s + 131-s + ⋯ |
| L(s) = 1 | + 2·4-s + 2/3·9-s + 1.80·11-s + 3·16-s − 0.458·19-s − 2.22·29-s − 1.43·31-s + 4/3·36-s − 1.87·41-s + 3.61·44-s + 13/7·49-s + 1.56·59-s − 0.256·61-s + 4·64-s + 1.42·71-s − 0.917·76-s − 1.80·79-s − 5/9·81-s − 2.54·89-s + 1.20·99-s + 1.19·101-s + 3.06·109-s − 4.45·116-s + 5/11·121-s − 2.87·124-s + 0.0887·127-s + 0.0873·131-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 225625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 225625 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.406578321\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.406578321\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 19 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 2 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 3 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 7 | $C_2^2$ | \( 1 - 13 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 - 25 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 85 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 85 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 38 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 118 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 97 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 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
−11.29120496476569501204698140684, −11.01599223829674273913371556322, −10.22769624515281842547939095267, −10.13558607609898692360040262003, −9.575684995004529791637221617645, −8.857929922065150000445823434414, −8.699800904324615898131129016567, −7.85569721733642044291143353334, −7.26522410276735085599114308423, −7.19338374257647227334403388816, −6.68555681052251664952406146848, −6.30420111822866218887948119414, −5.60467928863612811026237726981, −5.42193155285699158519675889024, −4.24437554745968537353075381868, −3.72626165952500761833576849287, −3.43417800691183099035068675520, −2.38918019682382824264125102261, −1.81850734941205597984963378329, −1.33790913205547663716745728777,
1.33790913205547663716745728777, 1.81850734941205597984963378329, 2.38918019682382824264125102261, 3.43417800691183099035068675520, 3.72626165952500761833576849287, 4.24437554745968537353075381868, 5.42193155285699158519675889024, 5.60467928863612811026237726981, 6.30420111822866218887948119414, 6.68555681052251664952406146848, 7.19338374257647227334403388816, 7.26522410276735085599114308423, 7.85569721733642044291143353334, 8.699800904324615898131129016567, 8.857929922065150000445823434414, 9.575684995004529791637221617645, 10.13558607609898692360040262003, 10.22769624515281842547939095267, 11.01599223829674273913371556322, 11.29120496476569501204698140684
