| L(s) = 1 | + 2·3-s − 4·5-s + 3·9-s − 4·11-s − 8·15-s − 12·17-s − 8·19-s − 4·23-s + 4·25-s + 4·27-s − 8·33-s + 8·37-s − 12·41-s − 12·45-s − 8·47-s − 24·51-s − 4·53-s + 16·55-s − 16·57-s − 8·61-s − 8·69-s − 4·71-s − 24·73-s + 8·75-s + 16·79-s + 5·81-s + 8·83-s + ⋯ |
| L(s) = 1 | + 1.15·3-s − 1.78·5-s + 9-s − 1.20·11-s − 2.06·15-s − 2.91·17-s − 1.83·19-s − 0.834·23-s + 4/5·25-s + 0.769·27-s − 1.39·33-s + 1.31·37-s − 1.87·41-s − 1.78·45-s − 1.16·47-s − 3.36·51-s − 0.549·53-s + 2.15·55-s − 2.11·57-s − 1.02·61-s − 0.963·69-s − 0.474·71-s − 2.80·73-s + 0.923·75-s + 1.80·79-s + 5/9·81-s + 0.878·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1382976 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1382976 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | | \( 1 \) |
| 3 | $C_1$ | \( ( 1 - T )^{2} \) |
| 7 | | \( 1 \) |
| good | 5 | $D_{4}$ | \( 1 + 4 T + 12 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + 4 T + 18 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 + 8 T^{2} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 12 T + 4 p T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 8 T + 46 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_4$ | \( 1 + 4 T + 42 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 50 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 + 54 T^{2} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 8 T + 58 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 12 T + 100 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 - 42 T^{2} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 8 T + 38 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 + 46 T^{2} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 8 T + 88 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 6 T^{2} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 4 T + 74 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 24 T + 272 T^{2} + 24 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 16 T + 190 T^{2} - 16 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 89 | $D_{4}$ | \( 1 + 20 T + 260 T^{2} + 20 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 8 T + 192 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
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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
−9.419303350709704968929559435681, −9.000102957278985376729721504384, −8.456954702600425368658508180777, −8.454906109540582547411570162117, −8.024401325852214141950400935372, −7.70845959287307296480670627157, −7.17407341273180023370414459577, −6.83580146482566967824414491633, −6.33368386325321103407459977393, −5.95682612880383446837192514269, −4.93556421804402001205414321792, −4.61181673767989521009336032026, −4.14655096086818676796755738349, −4.09524871302631246928186089105, −3.25458121864898608177438919053, −2.85110381088410268682063317892, −2.12671340534853567142878342682, −1.87775008412290576107459719475, 0, 0,
1.87775008412290576107459719475, 2.12671340534853567142878342682, 2.85110381088410268682063317892, 3.25458121864898608177438919053, 4.09524871302631246928186089105, 4.14655096086818676796755738349, 4.61181673767989521009336032026, 4.93556421804402001205414321792, 5.95682612880383446837192514269, 6.33368386325321103407459977393, 6.83580146482566967824414491633, 7.17407341273180023370414459577, 7.70845959287307296480670627157, 8.024401325852214141950400935372, 8.454906109540582547411570162117, 8.456954702600425368658508180777, 9.000102957278985376729721504384, 9.419303350709704968929559435681
