| L(s) = 1 | − 5·4-s + 290·9-s − 392·11-s − 523·16-s + 6.36e3·19-s + 7.84e3·29-s − 2.19e3·31-s − 1.45e3·36-s + 5.55e4·41-s + 1.96e3·44-s + 4.53e4·49-s − 2.39e4·59-s − 4.87e4·61-s + 235·64-s − 1.74e5·71-s − 3.18e4·76-s − 1.30e5·79-s + 4.13e4·81-s + 1.45e5·89-s − 1.13e5·99-s + 1.46e5·101-s + 4.59e5·109-s − 3.92e4·116-s − 2.46e5·121-s + 1.09e4·124-s + 127-s + 131-s + ⋯ |
| L(s) = 1 | − 0.156·4-s + 1.19·9-s − 0.976·11-s − 0.510·16-s + 4.04·19-s + 1.73·29-s − 0.409·31-s − 0.186·36-s + 5.15·41-s + 0.152·44-s + 2.69·49-s − 0.894·59-s − 1.67·61-s + 0.00717·64-s − 4.11·71-s − 0.631·76-s − 2.36·79-s + 0.700·81-s + 1.94·89-s − 1.16·99-s + 1.43·101-s + 3.70·109-s − 0.270·116-s − 1.53·121-s + 0.0640·124-s + 5.50e−6·127-s + 5.09e−6·131-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 390625 ^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 390625 ^{s/2} \, \Gamma_{\C}(s+5/2)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(3.195350494\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.195350494\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 5 | | \( 1 \) |
| good | 2 | $D_4\times C_2$ | \( 1 + 5 T^{2} + 137 p^{2} T^{4} + 5 p^{10} T^{6} + p^{20} T^{8} \) |
| 3 | $D_4\times C_2$ | \( 1 - 290 T^{2} + 4747 p^{2} T^{4} - 290 p^{10} T^{6} + p^{20} T^{8} \) |
| 7 | $D_4\times C_2$ | \( 1 - 45300 T^{2} + 1039412998 T^{4} - 45300 p^{10} T^{6} + p^{20} T^{8} \) |
| 11 | $D_{4}$ | \( ( 1 + 196 T + 181081 T^{2} + 196 p^{5} T^{3} + p^{10} T^{4} )^{2} \) |
| 13 | $D_4\times C_2$ | \( 1 - 1296980 T^{2} + 688260462198 T^{4} - 1296980 p^{10} T^{6} + p^{20} T^{8} \) |
| 17 | $D_4\times C_2$ | \( 1 - 2340610 T^{2} + 2927557675523 T^{4} - 2340610 p^{10} T^{6} + p^{20} T^{8} \) |
| 19 | $D_{4}$ | \( ( 1 - 3180 T + 7185073 T^{2} - 3180 p^{5} T^{3} + p^{10} T^{4} )^{2} \) |
| 23 | $D_4\times C_2$ | \( 1 - 24511220 T^{2} + 233031884985798 T^{4} - 24511220 p^{10} T^{6} + p^{20} T^{8} \) |
| 29 | $D_{4}$ | \( ( 1 - 3920 T + 44478298 T^{2} - 3920 p^{5} T^{3} + p^{10} T^{4} )^{2} \) |
| 31 | $D_{4}$ | \( ( 1 + 1096 T - 15343894 T^{2} + 1096 p^{5} T^{3} + p^{10} T^{4} )^{2} \) |
| 37 | $D_4\times C_2$ | \( 1 - 257567180 T^{2} + 26166130610514198 T^{4} - 257567180 p^{10} T^{6} + p^{20} T^{8} \) |
| 41 | $D_{4}$ | \( ( 1 - 27754 T + 414643531 T^{2} - 27754 p^{5} T^{3} + p^{10} T^{4} )^{2} \) |
| 43 | $D_4\times C_2$ | \( 1 - 37863500 T^{2} + 41125859560630998 T^{4} - 37863500 p^{10} T^{6} + p^{20} T^{8} \) |
| 47 | $D_4\times C_2$ | \( 1 - 298326940 T^{2} + 32136931855726598 T^{4} - 298326940 p^{10} T^{6} + p^{20} T^{8} \) |
| 53 | $D_4\times C_2$ | \( 1 - 1240678540 T^{2} + 709794326287792598 T^{4} - 1240678540 p^{10} T^{6} + p^{20} T^{8} \) |
| 59 | $D_{4}$ | \( ( 1 + 11960 T + 1234152598 T^{2} + 11960 p^{5} T^{3} + p^{10} T^{4} )^{2} \) |
| 61 | $D_{4}$ | \( ( 1 + 24396 T + 1596983806 T^{2} + 24396 p^{5} T^{3} + p^{10} T^{4} )^{2} \) |
| 67 | $D_4\times C_2$ | \( 1 - 4294993410 T^{2} + 8014205534576787523 T^{4} - 4294993410 p^{10} T^{6} + p^{20} T^{8} \) |
| 71 | $D_{4}$ | \( ( 1 + 87296 T + 5453356606 T^{2} + 87296 p^{5} T^{3} + p^{10} T^{4} )^{2} \) |
| 73 | $D_4\times C_2$ | \( 1 - 5672940770 T^{2} + 16272726164284201923 T^{4} - 5672940770 p^{10} T^{6} + p^{20} T^{8} \) |
| 79 | $D_{4}$ | \( ( 1 + 65480 T + 5707696298 T^{2} + 65480 p^{5} T^{3} + p^{10} T^{4} )^{2} \) |
| 83 | $D_4\times C_2$ | \( 1 - 11381926610 T^{2} + 63038986097658341523 T^{4} - 11381926610 p^{10} T^{6} + p^{20} T^{8} \) |
| 89 | $D_{4}$ | \( ( 1 - 72810 T + 5926578523 T^{2} - 72810 p^{5} T^{3} + p^{10} T^{4} )^{2} \) |
| 97 | $D_4\times C_2$ | \( 1 - 11961289340 T^{2} + 68433641741002238598 T^{4} - 11961289340 p^{10} T^{6} + p^{20} T^{8} \) |
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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
−12.14540939484042289857060710099, −11.90523044566214385587897925862, −11.49928751644936514407336830291, −11.33436925882232191672499237582, −10.57556128662513838306914408334, −10.39122002952179453712342248818, −10.20211745679126918043741198789, −9.672876579993069266008313221120, −9.258705529028324016959758688327, −9.036303311022754685227716307362, −8.764450992946443129523729325976, −7.72132452802252651792505504663, −7.48680807645462244794859444809, −7.48600928504577769048600575301, −7.26605538204975956496636512817, −6.13054596981145812317308703548, −5.96381000751118474725766750654, −5.44693637640829885635126766000, −4.76564772446388301793929903484, −4.50796753060133859929917454680, −3.84125762573755899005461780097, −2.84117052438160518418823621882, −2.73982120197850173732765105466, −1.26538541216834812006344965203, −0.830399145543979191803325939975,
0.830399145543979191803325939975, 1.26538541216834812006344965203, 2.73982120197850173732765105466, 2.84117052438160518418823621882, 3.84125762573755899005461780097, 4.50796753060133859929917454680, 4.76564772446388301793929903484, 5.44693637640829885635126766000, 5.96381000751118474725766750654, 6.13054596981145812317308703548, 7.26605538204975956496636512817, 7.48600928504577769048600575301, 7.48680807645462244794859444809, 7.72132452802252651792505504663, 8.764450992946443129523729325976, 9.036303311022754685227716307362, 9.258705529028324016959758688327, 9.672876579993069266008313221120, 10.20211745679126918043741198789, 10.39122002952179453712342248818, 10.57556128662513838306914408334, 11.33436925882232191672499237582, 11.49928751644936514407336830291, 11.90523044566214385587897925862, 12.14540939484042289857060710099
