| L(s) = 1 | − 92·5-s − 84·13-s − 1.92e3·17-s + 98·25-s + 5.10e3·29-s + 2.39e4·37-s + 1.01e4·41-s − 5.96e3·49-s + 3.94e4·53-s + 5.86e4·61-s + 7.72e3·65-s + 7.58e4·73-s + 1.77e5·85-s − 2.78e4·89-s + 3.27e5·97-s + 2.97e5·101-s + 2.46e5·109-s + 1.02e5·113-s − 3.15e5·121-s + 4.73e5·125-s + 127-s + 131-s + 137-s + 139-s − 4.69e5·145-s + 149-s + 151-s + ⋯ |
| L(s) = 1 | − 1.64·5-s − 0.137·13-s − 1.61·17-s + 0.0313·25-s + 1.12·29-s + 2.87·37-s + 0.943·41-s − 0.354·49-s + 1.92·53-s + 2.01·61-s + 0.226·65-s + 1.66·73-s + 2.65·85-s − 0.372·89-s + 3.53·97-s + 2.89·101-s + 1.98·109-s + 0.755·113-s − 1.95·121-s + 2.70·125-s + 5.50e−6·127-s + 5.09e−6·131-s + 4.55e−6·137-s + 4.38e−6·139-s − 1.85·145-s + 3.69e−6·149-s + 3.56e−6·151-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 82944 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 82944 ^{s/2} \, \Gamma_{\C}(s+5/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(1.759320850\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.759320850\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | | \( 1 \) |
| 3 | | \( 1 \) |
| good | 5 | $C_2$ | \( ( 1 + 46 T + p^{5} T^{2} )^{2} \) |
| 7 | $C_2^2$ | \( 1 + 5966 T^{2} + p^{10} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 315190 T^{2} + p^{10} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + 42 T + p^{5} T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 + 962 T + p^{5} T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( 1 + 632198 T^{2} + p^{10} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 2891758 T^{2} + p^{10} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 2554 T + p^{5} T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 + 53276990 T^{2} + p^{10} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - 11950 T + p^{5} T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 5078 T + p^{5} T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 + 136416374 T^{2} + p^{10} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 307289566 T^{2} + p^{10} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 19714 T + p^{5} T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 + 1350713110 T^{2} + p^{10} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 29318 T + p^{5} T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 + 2415413606 T^{2} + p^{10} T^{4} \) |
| 71 | $C_2^2$ | \( 1 - 2948789810 T^{2} + p^{10} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 37914 T + p^{5} T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 - 1729880290 T^{2} + p^{10} T^{4} \) |
| 83 | $C_2^2$ | \( 1 + 6331666438 T^{2} + p^{10} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 13930 T + p^{5} T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 163602 T + p^{5} T^{2} )^{2} \) |
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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
−11.34383389752881068094036698900, −10.99049925932865542019027855420, −10.08404055037443828895290050110, −10.03949827383007200008594110614, −9.044993264307684688384853875608, −8.920107024939429195089905065997, −8.120770306359113842152721198164, −7.948374890925111551907800457713, −7.37579898530281104887808030468, −6.94464611028993456769383353896, −6.22569251421134925622080811415, −5.91110597448325712623150697382, −4.70845915791628599869013445088, −4.69104981662115926391125119210, −3.85673858882470782456961428808, −3.64722337738717004396749739984, −2.51079726718942078248683489188, −2.24122681438354862994139369915, −0.818292125960851381695758030822, −0.51034730826459727123039658323,
0.51034730826459727123039658323, 0.818292125960851381695758030822, 2.24122681438354862994139369915, 2.51079726718942078248683489188, 3.64722337738717004396749739984, 3.85673858882470782456961428808, 4.69104981662115926391125119210, 4.70845915791628599869013445088, 5.91110597448325712623150697382, 6.22569251421134925622080811415, 6.94464611028993456769383353896, 7.37579898530281104887808030468, 7.948374890925111551907800457713, 8.120770306359113842152721198164, 8.920107024939429195089905065997, 9.044993264307684688384853875608, 10.03949827383007200008594110614, 10.08404055037443828895290050110, 10.99049925932865542019027855420, 11.34383389752881068094036698900
