| L(s) = 1 | + 242·4-s − 1.35e3·7-s − 6.16e4·13-s − 6.97e3·16-s − 2.76e5·19-s + 5.98e5·25-s − 3.28e5·28-s + 7.04e5·31-s + 2.37e6·37-s + 1.24e7·43-s − 1.01e7·49-s − 1.49e7·52-s + 3.31e7·61-s − 1.75e7·64-s + 1.53e7·67-s + 4.98e7·73-s − 6.69e7·76-s + 8.33e7·79-s + 8.36e7·91-s − 2.11e8·97-s + 1.44e8·100-s + 5.17e7·103-s − 2.19e8·109-s + 9.46e6·112-s + 2.49e8·121-s + 1.70e8·124-s + 127-s + ⋯ |
| L(s) = 1 | + 0.945·4-s − 0.565·7-s − 2.15·13-s − 0.106·16-s − 2.12·19-s + 1.53·25-s − 0.534·28-s + 0.762·31-s + 1.26·37-s + 3.65·43-s − 1.76·49-s − 2.03·52-s + 2.39·61-s − 1.04·64-s + 0.760·67-s + 1.75·73-s − 2.00·76-s + 2.14·79-s + 1.22·91-s − 2.39·97-s + 1.44·100-s + 0.459·103-s − 1.55·109-s + 0.0601·112-s + 1.16·121-s + 0.721·124-s + 1.20·133-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s+4)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{9}{2})\) |
\(\approx\) |
\(1.892186632\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.892186632\) |
| \(L(5)\) |
|
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 | | \( 1 \) |
| good | 2 | $C_2^2$ | \( 1 - 121 p T^{2} + p^{16} T^{4} \) |
| 5 | $C_2^2$ | \( 1 - 119746 p T^{2} + p^{16} T^{4} \) |
| 7 | $C_2$ | \( ( 1 + 97 p T + p^{8} T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( 1 - 2064602 p^{2} T^{2} + p^{16} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + 30817 T + p^{8} T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 + 2500566838 T^{2} + p^{16} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + 138391 T + p^{8} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 64394203082 T^{2} + p^{16} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 759623026078 T^{2} + p^{16} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 352214 T + p^{8} T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 1189991 T + p^{8} T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - 14772968430242 T^{2} + p^{16} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 6246086 T + p^{8} T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 47622567568202 T^{2} + p^{16} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 33684759113758 T^{2} + p^{16} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 182651688910922 T^{2} + p^{16} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 16580399 T + p^{8} T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 7667153 T + p^{8} T^{2} )^{2} \) |
| 71 | $C_2^2$ | \( 1 - 752286922698242 T^{2} + p^{16} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 24949631 T + p^{8} T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 41685089 T + p^{8} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 2503989590506082 T^{2} + p^{16} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 7872628964361482 T^{2} + p^{16} T^{4} \) |
| 97 | $C_2$ | \( ( 1 + 105926089 T + p^{8} T^{2} )^{2} \) |
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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
−15.99404548306247767802434295871, −14.96883344535685330261818307806, −14.84714222363863103811257621459, −14.16344382070777094656712276797, −13.07345585418015107305737895843, −12.39494114115470395758272949270, −12.37322608397139633357734425430, −11.02038796425741451751658987422, −10.93686406744189949888581782278, −9.874790776959160213759389323245, −9.387476464449158266887972907222, −8.359051456261112438527311289079, −7.49975358554839922812457170890, −6.74705960993782528510696168620, −6.30580835811470775177758640253, −5.02901659207401034809314560333, −4.18085328076651790579320775088, −2.55351047092598079542655566795, −2.39024521247924996975960912689, −0.59446233608563724080527499797,
0.59446233608563724080527499797, 2.39024521247924996975960912689, 2.55351047092598079542655566795, 4.18085328076651790579320775088, 5.02901659207401034809314560333, 6.30580835811470775177758640253, 6.74705960993782528510696168620, 7.49975358554839922812457170890, 8.359051456261112438527311289079, 9.387476464449158266887972907222, 9.874790776959160213759389323245, 10.93686406744189949888581782278, 11.02038796425741451751658987422, 12.37322608397139633357734425430, 12.39494114115470395758272949270, 13.07345585418015107305737895843, 14.16344382070777094656712276797, 14.84714222363863103811257621459, 14.96883344535685330261818307806, 15.99404548306247767802434295871
