| L(s) = 1 | + 4-s + 2·7-s + 16-s − 4·25-s + 2·28-s − 20·37-s + 16·43-s − 3·49-s + 64-s − 8·67-s − 20·79-s − 4·100-s + 28·109-s + 2·112-s + 121-s + 127-s + 131-s + 137-s + 139-s − 20·148-s + 149-s + 151-s + 157-s + 163-s + 167-s + 20·169-s + 16·172-s + ⋯ |
| L(s) = 1 | + 1/2·4-s + 0.755·7-s + 1/4·16-s − 4/5·25-s + 0.377·28-s − 3.28·37-s + 2.43·43-s − 3/7·49-s + 1/8·64-s − 0.977·67-s − 2.25·79-s − 2/5·100-s + 2.68·109-s + 0.188·112-s + 1/11·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 1.64·148-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 1.53·169-s + 1.21·172-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1920996 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1920996 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 3 | | \( 1 \) |
| 7 | $C_2$ | \( 1 - 2 T + p T^{2} \) |
| 11 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| good | 5 | $C_2^2$ | \( 1 + 4 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 - 20 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( 1 + 16 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - 14 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 59 | $C_2^2$ | \( 1 + 112 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 68 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 122 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 + 160 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 + 82 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 - 170 T^{2} + p^{2} T^{4} \) |
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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
−7.56311450479711011223892830063, −7.16671454717837988658495734408, −6.87119209108885998796730867702, −6.18658308120414096161997514241, −5.87427409552722057966243886460, −5.47571428639151072915653297005, −4.90146344831662060345765604480, −4.60517896303743097493674670424, −3.85158907943874317142561914692, −3.61418254472565950027747673009, −2.87455712229305928948377486660, −2.34028424867006508059765505119, −1.73721351772154855694667955760, −1.24739216296672214036118850140, 0,
1.24739216296672214036118850140, 1.73721351772154855694667955760, 2.34028424867006508059765505119, 2.87455712229305928948377486660, 3.61418254472565950027747673009, 3.85158907943874317142561914692, 4.60517896303743097493674670424, 4.90146344831662060345765604480, 5.47571428639151072915653297005, 5.87427409552722057966243886460, 6.18658308120414096161997514241, 6.87119209108885998796730867702, 7.16671454717837988658495734408, 7.56311450479711011223892830063
