| L(s) = 1 | + 2.90e3·9-s + 4.56e3·11-s − 4.19e4·19-s + 4.61e5·29-s + 4.29e5·31-s + 8.98e5·41-s + 3.07e6·49-s + 1.88e6·59-s + 2.53e6·61-s + 7.23e6·71-s + 6.07e6·79-s + 8.01e5·81-s − 6.47e6·89-s + 1.32e7·99-s − 4.67e7·101-s − 5.67e7·109-s − 5.47e7·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 1.13e8·169-s + ⋯ |
| L(s) = 1 | + 1.32·9-s + 1.03·11-s − 1.40·19-s + 3.51·29-s + 2.58·31-s + 2.03·41-s + 3.73·49-s + 1.19·59-s + 1.42·61-s + 2.39·71-s + 1.38·79-s + 0.167·81-s − 0.974·89-s + 1.37·99-s − 4.51·101-s − 4.19·109-s − 2.80·121-s + 1.81·169-s − 1.86·171-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(8-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s+7/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(10.43503526\) |
| \(L(\frac12)\) |
\(\approx\) |
\(10.43503526\) |
| \(L(\frac{9}{2})\) |
|
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 | | \( 1 \) |
| 5 | | \( 1 \) |
| good | 3 | $D_4\times C_2$ | \( 1 - 2906 T^{2} + 849283 p^{2} T^{4} - 2906 p^{14} T^{6} + p^{28} T^{8} \) |
| 7 | $D_4\times C_2$ | \( 1 - 3073460 T^{2} + 3710869968198 T^{4} - 3073460 p^{14} T^{6} + p^{28} T^{8} \) |
| 11 | $D_{4}$ | \( ( 1 - 2280 T + 35168917 T^{2} - 2280 p^{7} T^{3} + p^{14} T^{4} )^{2} \) |
| 13 | $D_4\times C_2$ | \( 1 - 113927156 T^{2} + 9357010691503062 T^{4} - 113927156 p^{14} T^{6} + p^{28} T^{8} \) |
| 17 | $D_4\times C_2$ | \( 1 - 276118034 T^{2} - 43412344711209453 T^{4} - 276118034 p^{14} T^{6} + p^{28} T^{8} \) |
| 19 | $D_{4}$ | \( ( 1 + 20968 T - 76852491 T^{2} + 20968 p^{7} T^{3} + p^{14} T^{4} )^{2} \) |
| 23 | $D_4\times C_2$ | \( 1 - 12837939860 T^{2} + 64344944571444482118 T^{4} - 12837939860 p^{14} T^{6} + p^{28} T^{8} \) |
| 29 | $D_{4}$ | \( ( 1 - 230952 T + 37951130794 T^{2} - 230952 p^{7} T^{3} + p^{14} T^{4} )^{2} \) |
| 31 | $D_{4}$ | \( ( 1 - 214672 T + 66489522618 T^{2} - 214672 p^{7} T^{3} + p^{14} T^{4} )^{2} \) |
| 37 | $D_4\times C_2$ | \( 1 - 78783805580 T^{2} + \)\(11\!\cdots\!78\)\( T^{4} - 78783805580 p^{14} T^{6} + p^{28} T^{8} \) |
| 41 | $D_{4}$ | \( ( 1 - 449082 T + 267091482043 T^{2} - 449082 p^{7} T^{3} + p^{14} T^{4} )^{2} \) |
| 43 | $D_4\times C_2$ | \( 1 - 501512069228 T^{2} + \)\(12\!\cdots\!94\)\( T^{4} - 501512069228 p^{14} T^{6} + p^{28} T^{8} \) |
| 47 | $D_4\times C_2$ | \( 1 + 32460472036 T^{2} + \)\(30\!\cdots\!62\)\( T^{4} + 32460472036 p^{14} T^{6} + p^{28} T^{8} \) |
| 53 | $D_4\times C_2$ | \( 1 - 1901262944300 T^{2} + \)\(33\!\cdots\!38\)\( T^{4} - 1901262944300 p^{14} T^{6} + p^{28} T^{8} \) |
| 59 | $D_{4}$ | \( ( 1 - 943824 T + 1101608154982 T^{2} - 943824 p^{7} T^{3} + p^{14} T^{4} )^{2} \) |
| 61 | $D_{4}$ | \( ( 1 - 1266364 T + 5241560521566 T^{2} - 1266364 p^{7} T^{3} + p^{14} T^{4} )^{2} \) |
| 67 | $D_4\times C_2$ | \( 1 - 19031107681610 T^{2} + \)\(16\!\cdots\!83\)\( T^{4} - 19031107681610 p^{14} T^{6} + p^{28} T^{8} \) |
| 71 | $D_{4}$ | \( ( 1 - 3616776 T + 18754925905726 T^{2} - 3616776 p^{7} T^{3} + p^{14} T^{4} )^{2} \) |
| 73 | $D_4\times C_2$ | \( 1 - 38620847375186 T^{2} + \)\(61\!\cdots\!67\)\( T^{4} - 38620847375186 p^{14} T^{6} + p^{28} T^{8} \) |
| 79 | $D_{4}$ | \( ( 1 - 3039896 T + 32690326402122 T^{2} - 3039896 p^{7} T^{3} + p^{14} T^{4} )^{2} \) |
| 83 | $D_4\times C_2$ | \( 1 - 78938156700890 T^{2} + \)\(30\!\cdots\!83\)\( T^{4} - 78938156700890 p^{14} T^{6} + p^{28} T^{8} \) |
| 89 | $D_{4}$ | \( ( 1 + 3239262 T + 52422947109619 T^{2} + 3239262 p^{7} T^{3} + p^{14} T^{4} )^{2} \) |
| 97 | $D_4\times C_2$ | \( 1 - 219098922513980 T^{2} + \)\(23\!\cdots\!38\)\( T^{4} - 219098922513980 p^{14} T^{6} + p^{28} T^{8} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.905665530859348128249231178394, −8.238018398450104144272287713822, −8.162429220092981517851264039213, −8.151869695063773127171690208957, −7.77205504681082945705525273062, −6.98671699207895459958325724265, −6.88655270283610705555230968723, −6.63334418952088024923470982110, −6.60741416771688818222340767291, −6.26231397104902268152556656255, −5.56083100393305491228776021785, −5.42945192839853483389106787489, −5.07254699168319736828607604672, −4.29695372555632232045403028506, −4.20824379326115214589073662704, −4.16300282544087449269012107293, −4.02246041591950156945331325700, −2.91311881297775938636821986480, −2.89594248181598497924728784597, −2.28848962304331392235233487491, −2.24222664607416326600541455451, −1.33310220991261504048608769809, −0.962874104318930371615306893208, −0.941813800632492313637483314191, −0.49363155026643849715233893843,
0.49363155026643849715233893843, 0.941813800632492313637483314191, 0.962874104318930371615306893208, 1.33310220991261504048608769809, 2.24222664607416326600541455451, 2.28848962304331392235233487491, 2.89594248181598497924728784597, 2.91311881297775938636821986480, 4.02246041591950156945331325700, 4.16300282544087449269012107293, 4.20824379326115214589073662704, 4.29695372555632232045403028506, 5.07254699168319736828607604672, 5.42945192839853483389106787489, 5.56083100393305491228776021785, 6.26231397104902268152556656255, 6.60741416771688818222340767291, 6.63334418952088024923470982110, 6.88655270283610705555230968723, 6.98671699207895459958325724265, 7.77205504681082945705525273062, 8.151869695063773127171690208957, 8.162429220092981517851264039213, 8.238018398450104144272287713822, 8.905665530859348128249231178394
