| L(s) = 1 | − 18·3-s − 16·4-s + 48·5-s + 243·9-s + 288·12-s − 864·15-s − 768·16-s − 768·20-s + 780·23-s − 4.52e3·25-s − 2.91e3·27-s − 4.78e3·31-s − 3.88e3·36-s + 1.02e4·37-s + 1.16e4·45-s − 2.65e4·47-s + 1.38e4·48-s − 3.36e4·49-s − 468·53-s + 7.62e4·59-s + 1.38e4·60-s + 2.86e4·64-s + 5.86e4·67-s − 1.40e4·69-s − 1.46e5·71-s + 8.13e4·75-s − 3.68e4·80-s + ⋯ |
| L(s) = 1 | − 1.15·3-s − 1/2·4-s + 0.858·5-s + 9-s + 0.577·12-s − 0.991·15-s − 3/4·16-s − 0.429·20-s + 0.307·23-s − 1.44·25-s − 0.769·27-s − 0.894·31-s − 1/2·36-s + 1.23·37-s + 0.858·45-s − 1.75·47-s + 0.866·48-s − 1.99·49-s − 0.0228·53-s + 2.85·59-s + 0.495·60-s + 7/8·64-s + 1.59·67-s − 0.355·69-s − 3.44·71-s + 1.67·75-s − 0.643·80-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 131769 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 131769 ^{s/2} \, \Gamma_{\C}(s+5/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | $C_1$ | \( ( 1 + p^{2} T )^{2} \) |
| 11 | | \( 1 \) |
| good | 2 | $C_2^2$ | \( 1 + p^{4} T^{2} + p^{10} T^{4} \) |
| 5 | $C_2$ | \( ( 1 - 24 T + p^{5} T^{2} )^{2} \) |
| 7 | $C_2^2$ | \( 1 + 33611 T^{2} + p^{10} T^{4} \) |
| 13 | $C_2^2$ | \( 1 + 693434 T^{2} + p^{10} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 995746 T^{2} + p^{10} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 2870531 T^{2} + p^{10} T^{4} \) |
| 23 | $C_2$ | \( ( 1 - 390 T + p^{5} T^{2} )^{2} \) |
| 29 | $C_2^2$ | \( 1 + 29847598 T^{2} + p^{10} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + 2393 T + p^{5} T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 5137 T + p^{5} T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - 39151610 T^{2} + p^{10} T^{4} \) |
| 43 | $C_2^2$ | \( 1 + 258722186 T^{2} + p^{10} T^{4} \) |
| 47 | $C_2$ | \( ( 1 + 13266 T + p^{5} T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 + 234 T + p^{5} T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 38118 T + p^{5} T^{2} )^{2} \) |
| 61 | $C_2^2$ | \( 1 + 888695927 T^{2} + p^{10} T^{4} \) |
| 67 | $C_2$ | \( ( 1 - 29335 T + p^{5} T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + 73212 T + p^{5} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 885792889 T^{2} + p^{10} T^{4} \) |
| 79 | $C_2^2$ | \( 1 + 5723623475 T^{2} + p^{10} T^{4} \) |
| 83 | $C_2^2$ | \( 1 + 2133793594 T^{2} + p^{10} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 98892 T + p^{5} T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 18485 T + p^{5} T^{2} )^{2} \) |
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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
−10.28937075428923477672728166587, −9.885050484450945930746122174304, −9.453268318410010979479634631068, −9.390934893558944413716637011411, −8.384693365992565184375469864871, −8.186587184768304547150633383894, −7.42275823229404829422310776381, −6.83127097239777994714848275621, −6.52955606596514415424310874464, −5.93472616512508984980816465352, −5.33137130829502182738626576055, −5.29718673939392752213693034930, −4.23737953008870060387635160760, −4.22194379145497791091277410615, −3.21240964846488017826907063839, −2.36630106325139942802669147056, −1.71550949962863224925829870473, −1.14167145851187611235123405662, 0, 0,
1.14167145851187611235123405662, 1.71550949962863224925829870473, 2.36630106325139942802669147056, 3.21240964846488017826907063839, 4.22194379145497791091277410615, 4.23737953008870060387635160760, 5.29718673939392752213693034930, 5.33137130829502182738626576055, 5.93472616512508984980816465352, 6.52955606596514415424310874464, 6.83127097239777994714848275621, 7.42275823229404829422310776381, 8.186587184768304547150633383894, 8.384693365992565184375469864871, 9.390934893558944413716637011411, 9.453268318410010979479634631068, 9.885050484450945930746122174304, 10.28937075428923477672728166587
