| L(s) = 1 | − 4·2-s − 52·4-s + 352·8-s + 138·9-s − 568·11-s + 1.61e3·16-s − 552·18-s + 2.27e3·22-s + 2.99e3·23-s − 634·25-s − 8.73e3·29-s − 1.96e4·32-s − 7.17e3·36-s − 2.52e4·37-s − 2.71e3·43-s + 2.95e4·44-s − 1.19e4·46-s + 2.53e3·50-s + 2.83e4·53-s + 3.49e4·58-s − 2.31e4·64-s − 7.28e3·67-s + 7.12e4·71-s + 4.85e4·72-s + 1.01e5·74-s − 1.09e5·79-s − 4.00e4·81-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.62·4-s + 1.94·8-s + 0.567·9-s − 1.41·11-s + 1.57·16-s − 0.401·18-s + 1.00·22-s + 1.17·23-s − 0.202·25-s − 1.92·29-s − 3.39·32-s − 0.922·36-s − 3.03·37-s − 0.223·43-s + 2.29·44-s − 0.833·46-s + 0.143·50-s + 1.38·53-s + 1.36·58-s − 0.705·64-s − 0.198·67-s + 1.67·71-s + 1.10·72-s + 2.14·74-s − 1.96·79-s − 0.677·81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2401 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2401 ^{s/2} \, \Gamma_{\C}(s+5/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.4408465064\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4408465064\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 7 | | \( 1 \) |
| good | 2 | $C_2$ | \( ( 1 + p T + p^{5} T^{2} )^{2} \) |
| 3 | $C_2^2$ | \( 1 - 46 p T^{2} + p^{10} T^{4} \) |
| 5 | $C_2^2$ | \( 1 + 634 T^{2} + p^{10} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + 284 T + p^{5} T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 + 35954 p T^{2} + p^{10} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 2817250 T^{2} + p^{10} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 229142 T^{2} + p^{10} T^{4} \) |
| 23 | $C_2$ | \( ( 1 - 1496 T + p^{5} T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + 4366 T + p^{5} T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 + 15722366 T^{2} + p^{10} T^{4} \) |
| 37 | $C_2$ | \( ( 1 + 12630 T + p^{5} T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 + 142552786 T^{2} + p^{10} T^{4} \) |
| 43 | $C_2$ | \( ( 1 + 1356 T + p^{5} T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 + 357849118 T^{2} + p^{10} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 14150 T + p^{5} T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 + 31458982 T^{2} + p^{10} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 422082602 T^{2} + p^{10} T^{4} \) |
| 67 | $C_2$ | \( ( 1 + 3644 T + p^{5} T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 35632 T + p^{5} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 + 2484166610 T^{2} + p^{10} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 54616 T + p^{5} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 + 7877806102 T^{2} + p^{10} T^{4} \) |
| 89 | $C_2^2$ | \( 1 + 10752624754 T^{2} + p^{10} T^{4} \) |
| 97 | $C_2^2$ | \( 1 - 16535359486 T^{2} + p^{10} T^{4} \) |
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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
−14.87666734531662154492780887845, −14.19011838591773733281930931658, −13.64143055369740617567370190390, −13.00774088918867840392777419336, −12.99491928948682677511876178260, −12.16452426120625643001517762712, −11.07654606259343191825190591051, −10.45225962195148426676160884157, −10.11000278058710878791113209938, −9.412060224854830657200647054635, −8.817329721601413741160473146994, −8.350377717238574837784015840520, −7.53713322833979450129283956013, −7.10458692726977830579516010790, −5.36834838610195200544384892644, −5.27889276202149891284105192365, −4.23966316132992123864667505701, −3.39528302569368729617318245128, −1.69132044578871428722533862179, −0.40630917411279454399531091558,
0.40630917411279454399531091558, 1.69132044578871428722533862179, 3.39528302569368729617318245128, 4.23966316132992123864667505701, 5.27889276202149891284105192365, 5.36834838610195200544384892644, 7.10458692726977830579516010790, 7.53713322833979450129283956013, 8.350377717238574837784015840520, 8.817329721601413741160473146994, 9.412060224854830657200647054635, 10.11000278058710878791113209938, 10.45225962195148426676160884157, 11.07654606259343191825190591051, 12.16452426120625643001517762712, 12.99491928948682677511876178260, 13.00774088918867840392777419336, 13.64143055369740617567370190390, 14.19011838591773733281930931658, 14.87666734531662154492780887845
