L(s) = 1 | + 2·3-s + 2·7-s − 4·11-s − 8·19-s + 4·21-s + 2·23-s − 2·27-s + 4·31-s − 8·33-s − 4·37-s + 4·41-s + 14·43-s + 10·47-s − 8·49-s + 16·53-s − 16·57-s − 8·59-s − 4·61-s + 18·67-s + 4·69-s + 4·71-s + 8·73-s − 8·77-s + 16·79-s − 81-s + 6·83-s − 4·89-s + ⋯ |
L(s) = 1 | + 1.15·3-s + 0.755·7-s − 1.20·11-s − 1.83·19-s + 0.872·21-s + 0.417·23-s − 0.384·27-s + 0.718·31-s − 1.39·33-s − 0.657·37-s + 0.624·41-s + 2.13·43-s + 1.45·47-s − 8/7·49-s + 2.19·53-s − 2.11·57-s − 1.04·59-s − 0.512·61-s + 2.19·67-s + 0.481·69-s + 0.474·71-s + 0.936·73-s − 0.911·77-s + 1.80·79-s − 1/9·81-s + 0.658·83-s − 0.423·89-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 40960000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 40960000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(4.315601350\) |
\(L(\frac12)\) |
\(\approx\) |
\(4.315601350\) |
\(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | | \( 1 \) |
good | 3 | $D_{4}$ | \( 1 - 2 T + 4 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 7 | $D_{4}$ | \( 1 - 2 T + 12 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 22 T^{2} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 8 T + 42 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 2 T + 20 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 4 T + 54 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 41 | $D_{4}$ | \( 1 - 4 T + 74 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 14 T + 132 T^{2} - 14 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 10 T + 116 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 16 T + 158 T^{2} - 16 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 8 T + 122 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 4 T + 78 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 18 T + 212 T^{2} - 18 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 4 T + 134 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 8 T + 150 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 16 T + 174 T^{2} - 16 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 6 T + 172 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 4 T + 134 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 8 T + 102 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
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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
−8.397670926496451013342636088344, −7.86443143344596995666334874393, −7.57660033119691154512905803500, −7.41884357180499742019760399306, −6.78813382643303652776061422883, −6.52951589882592958637023999489, −6.06208008853187661074165870693, −5.70565603300250675554204431808, −5.31514525896768511393669883743, −4.95478009257239209775591657031, −4.51127310565818569944930997220, −4.25570923142634399953283020848, −3.74519012339794992906325946262, −3.39136501004962836438491469752, −2.82470895842994540170339506858, −2.50432464151061273289451744450, −2.10874670994561817983178928237, −2.00815963169906353922911419094, −0.949551046174372709247966106175, −0.54436499459371984814102479984,
0.54436499459371984814102479984, 0.949551046174372709247966106175, 2.00815963169906353922911419094, 2.10874670994561817983178928237, 2.50432464151061273289451744450, 2.82470895842994540170339506858, 3.39136501004962836438491469752, 3.74519012339794992906325946262, 4.25570923142634399953283020848, 4.51127310565818569944930997220, 4.95478009257239209775591657031, 5.31514525896768511393669883743, 5.70565603300250675554204431808, 6.06208008853187661074165870693, 6.52951589882592958637023999489, 6.78813382643303652776061422883, 7.41884357180499742019760399306, 7.57660033119691154512905803500, 7.86443143344596995666334874393, 8.397670926496451013342636088344