| L(s) = 1 | + 54·3-s + 1.32e3·4-s − 3.44e4·7-s − 5.61e4·9-s + 7.17e4·12-s + 3.39e5·13-s + 7.15e5·16-s − 1.89e6·19-s − 1.86e6·21-s − 6.21e6·27-s − 4.57e7·28-s − 5.95e7·31-s − 7.45e7·36-s + 1.21e8·37-s + 1.83e7·39-s − 2.14e8·43-s + 3.86e7·48-s + 3.26e8·49-s + 4.50e8·52-s − 1.02e8·57-s + 2.06e9·61-s + 1.93e9·63-s − 4.42e8·64-s − 3.75e9·67-s + 5.69e9·73-s − 2.52e9·76-s + 2.97e9·79-s + ⋯ |
| L(s) = 1 | + 2/9·3-s + 1.29·4-s − 2.05·7-s − 0.950·9-s + 0.288·12-s + 0.913·13-s + 0.681·16-s − 0.766·19-s − 0.455·21-s − 0.433·27-s − 2.65·28-s − 2.08·31-s − 1.23·36-s + 1.75·37-s + 0.203·39-s − 1.46·43-s + 0.151·48-s + 1.15·49-s + 1.18·52-s − 0.170·57-s + 2.44·61-s + 1.94·63-s − 0.412·64-s − 2.78·67-s + 2.74·73-s − 0.994·76-s + 0.967·79-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(11-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5625 ^{s/2} \, \Gamma_{\C}(s+5)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{11}{2})\) |
\(\approx\) |
\(0.6791764386\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6791764386\) |
| \(L(6)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | $C_2$ | \( 1 - 2 p^{3} T + p^{10} T^{2} \) |
| 5 | | \( 1 \) |
| good | 2 | $C_2^2$ | \( 1 - 83 p^{4} T^{2} + p^{20} T^{4} \) |
| 7 | $C_2$ | \( ( 1 + 2462 p T + p^{10} T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( 1 - 16976849522 T^{2} + p^{20} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 169654 T + p^{10} T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 - 3914170410818 T^{2} + p^{20} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + 949462 T + p^{10} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 75790028393378 T^{2} + p^{20} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 831288000078482 T^{2} + p^{20} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + 29793118 T + p^{10} T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 60811846 T + p^{10} T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 + 6157996032931678 T^{2} + p^{20} T^{4} \) |
| 43 | $C_2$ | \( ( 1 + 107419706 T + p^{10} T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 33376598707262018 T^{2} + p^{20} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 312876100791567218 T^{2} + p^{20} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 600827685707033522 T^{2} + p^{20} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 1030793642 T + p^{10} T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 + 1876742474 T + p^{10} T^{2} )^{2} \) |
| 71 | $C_2^2$ | \( 1 + 690030731290713118 T^{2} + p^{20} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 2846528494 T + p^{10} T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 1488647618 T + p^{10} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 29429698146400299218 T^{2} + p^{20} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 26116812713945754722 T^{2} + p^{20} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 1592948926 T + p^{10} T^{2} )^{2} \) |
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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
−13.54959223186512424060185205908, −12.14882353933100213857803926077, −11.61263932897688874997950335025, −10.90787009442843262806232068970, −10.82456193546750703721563140697, −9.853505588414889804976270994659, −9.455657275808902767499387484111, −8.829079527934998726176771076563, −8.180473945292139535213087113731, −7.43310258240658460320348621354, −6.72827549160284585423709266379, −6.30377465623592879309485391155, −6.02318573629767536809167614018, −5.16529740974093405448420260532, −3.71468468022257196769335335557, −3.55619794876606857637096269018, −2.69082285223682535925976184184, −2.29497591530734920516895610784, −1.31704849685894953234116161337, −0.20862480879890466730974713464,
0.20862480879890466730974713464, 1.31704849685894953234116161337, 2.29497591530734920516895610784, 2.69082285223682535925976184184, 3.55619794876606857637096269018, 3.71468468022257196769335335557, 5.16529740974093405448420260532, 6.02318573629767536809167614018, 6.30377465623592879309485391155, 6.72827549160284585423709266379, 7.43310258240658460320348621354, 8.180473945292139535213087113731, 8.829079527934998726176771076563, 9.455657275808902767499387484111, 9.853505588414889804976270994659, 10.82456193546750703721563140697, 10.90787009442843262806232068970, 11.61263932897688874997950335025, 12.14882353933100213857803926077, 13.54959223186512424060185205908
