| L(s) = 1 | + 48·4-s − 162·9-s − 24·11-s + 796·16-s − 8.42e3·19-s − 8.13e3·29-s − 5.19e3·31-s − 7.77e3·36-s + 2.24e4·41-s − 1.15e3·44-s + 2.63e4·49-s + 1.27e5·59-s + 1.46e4·61-s + 1.49e4·64-s + 1.96e5·71-s − 4.04e5·76-s − 1.68e5·79-s + 1.96e4·81-s − 2.06e5·89-s + 3.88e3·99-s + 2.35e5·101-s − 4.03e5·109-s − 3.90e5·116-s − 1.63e5·121-s − 2.49e5·124-s + 127-s + 131-s + ⋯ |
| L(s) = 1 | + 3/2·4-s − 2/3·9-s − 0.0598·11-s + 0.777·16-s − 5.35·19-s − 1.79·29-s − 0.971·31-s − 36-s + 2.08·41-s − 0.0897·44-s + 1.56·49-s + 4.78·59-s + 0.503·61-s + 0.457·64-s + 4.62·71-s − 8.03·76-s − 3.02·79-s + 1/3·81-s − 2.75·89-s + 0.0398·99-s + 2.29·101-s − 3.25·109-s − 2.69·116-s − 1.01·121-s − 1.45·124-s + 5.50e−6·127-s + 5.09e−6·131-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 31640625 ^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 31640625 ^{s/2} \, \Gamma_{\C}(s+5/2)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.8992953244\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8992953244\) |
| \(L(\frac{7}{2})\) |
|
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 | $C_2$ | \( ( 1 + p^{4} T^{2} )^{2} \) |
| 5 | | \( 1 \) |
| good | 2 | $D_4\times C_2$ | \( 1 - 3 p^{4} T^{2} + 377 p^{2} T^{4} - 3 p^{14} T^{6} + p^{20} T^{8} \) |
| 7 | $D_4\times C_2$ | \( 1 - 26314 T^{2} + 552283323 T^{4} - 26314 p^{10} T^{6} + p^{20} T^{8} \) |
| 11 | $D_{4}$ | \( ( 1 + 12 T + 82074 T^{2} + 12 p^{5} T^{3} + p^{10} T^{4} )^{2} \) |
| 13 | $D_4\times C_2$ | \( 1 - 358322 T^{2} - 9568609917 T^{4} - 358322 p^{10} T^{6} + p^{20} T^{8} \) |
| 17 | $D_4\times C_2$ | \( 1 - 4107900 T^{2} + 8104643465798 T^{4} - 4107900 p^{10} T^{6} + p^{20} T^{8} \) |
| 19 | $D_{4}$ | \( ( 1 + 4214 T + 8516703 T^{2} + 4214 p^{5} T^{3} + p^{10} T^{4} )^{2} \) |
| 23 | $D_4\times C_2$ | \( 1 + 387244 T^{2} + 80324387862918 T^{4} + 387244 p^{10} T^{6} + p^{20} T^{8} \) |
| 29 | $D_{4}$ | \( ( 1 + 4068 T + 45062238 T^{2} + 4068 p^{5} T^{3} + p^{10} T^{4} )^{2} \) |
| 31 | $D_{4}$ | \( ( 1 + 2598 T + 31786727 T^{2} + 2598 p^{5} T^{3} + p^{10} T^{4} )^{2} \) |
| 37 | $D_4\times C_2$ | \( 1 - 72084044 T^{2} + 9012846235728918 T^{4} - 72084044 p^{10} T^{6} + p^{20} T^{8} \) |
| 41 | $D_{4}$ | \( ( 1 - 11232 T + 260535762 T^{2} - 11232 p^{5} T^{3} + p^{10} T^{4} )^{2} \) |
| 43 | $D_4\times C_2$ | \( 1 - 549439850 T^{2} + 118590729812334123 T^{4} - 549439850 p^{10} T^{6} + p^{20} T^{8} \) |
| 47 | $D_4\times C_2$ | \( 1 - 227668020 T^{2} + 116078671176173798 T^{4} - 227668020 p^{10} T^{6} + p^{20} T^{8} \) |
| 53 | $D_4\times C_2$ | \( 1 + 447693324 T^{2} + 391810911498356438 T^{4} + 447693324 p^{10} T^{6} + p^{20} T^{8} \) |
| 59 | $D_{4}$ | \( ( 1 - 63924 T + 2170928058 T^{2} - 63924 p^{5} T^{3} + p^{10} T^{4} )^{2} \) |
| 61 | $D_{4}$ | \( ( 1 - 7310 T + 1671053643 T^{2} - 7310 p^{5} T^{3} + p^{10} T^{4} )^{2} \) |
| 67 | $D_4\times C_2$ | \( 1 - 1844169850 T^{2} + 1350269600568616923 T^{4} - 1844169850 p^{10} T^{6} + p^{20} T^{8} \) |
| 71 | $D_{4}$ | \( ( 1 - 98304 T + 4832737806 T^{2} - 98304 p^{5} T^{3} + p^{10} T^{4} )^{2} \) |
| 73 | $D_4\times C_2$ | \( 1 - 7219366556 T^{2} + 21370998290339231718 T^{4} - 7219366556 p^{10} T^{6} + p^{20} T^{8} \) |
| 79 | $D_{4}$ | \( ( 1 + 84000 T + 7803834398 T^{2} + 84000 p^{5} T^{3} + p^{10} T^{4} )^{2} \) |
| 83 | $D_4\times C_2$ | \( 1 - 13556615492 T^{2} + 76898440141363796118 T^{4} - 13556615492 p^{10} T^{6} + p^{20} T^{8} \) |
| 89 | $D_{4}$ | \( ( 1 + 103104 T + 13514604658 T^{2} + 103104 p^{5} T^{3} + p^{10} T^{4} )^{2} \) |
| 97 | $D_4\times C_2$ | \( 1 - 1716348290 T^{2} + 75484636759946814723 T^{4} - 1716348290 p^{10} T^{6} + p^{20} T^{8} \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.972408841194751764995532269601, −9.266780657051289121541253439957, −9.257473812694577760063221088604, −8.627934711767634887549344596939, −8.552213463817247691166455528067, −8.187654119234261913745903875835, −8.133142058720669422244819190032, −7.44132946233210824455980734821, −7.04941335527330472046563810249, −6.88348096347018902803697400613, −6.62959645733080563066387657748, −6.36204390288129819746775266429, −5.76540100568683162364143862283, −5.68567265655657515173609931533, −5.40721661572317699295914213165, −4.61253316561686101495814455653, −4.12614886230068863629705394518, −3.89356210477527290674510296136, −3.75360533793660498221717968112, −2.61803303654815319968948993198, −2.42034942053725838000141087290, −2.05498408948762226439403784429, −2.04739751849434929427622546096, −0.918430641812776591865932449752, −0.18689518056172816197187734128,
0.18689518056172816197187734128, 0.918430641812776591865932449752, 2.04739751849434929427622546096, 2.05498408948762226439403784429, 2.42034942053725838000141087290, 2.61803303654815319968948993198, 3.75360533793660498221717968112, 3.89356210477527290674510296136, 4.12614886230068863629705394518, 4.61253316561686101495814455653, 5.40721661572317699295914213165, 5.68567265655657515173609931533, 5.76540100568683162364143862283, 6.36204390288129819746775266429, 6.62959645733080563066387657748, 6.88348096347018902803697400613, 7.04941335527330472046563810249, 7.44132946233210824455980734821, 8.133142058720669422244819190032, 8.187654119234261913745903875835, 8.552213463817247691166455528067, 8.627934711767634887549344596939, 9.257473812694577760063221088604, 9.266780657051289121541253439957, 9.972408841194751764995532269601
