| L(s) = 1 | + 2·2-s + 18·3-s + 18·4-s + 36·6-s + 28·7-s + 36·8-s + 189·9-s − 190·11-s + 324·12-s + 56·14-s + 264·16-s − 402·17-s + 378·18-s − 1.31e3·19-s + 504·21-s − 380·22-s + 110·23-s + 648·24-s + 125·25-s + 1.45e3·27-s + 504·28-s − 2.88e3·29-s + 3.49e3·31-s + 72·32-s − 3.42e3·33-s − 804·34-s + 3.40e3·36-s + ⋯ |
| L(s) = 1 | + 1/2·2-s + 2·3-s + 9/8·4-s + 6-s + 4/7·7-s + 9/16·8-s + 7/3·9-s − 1.57·11-s + 9/4·12-s + 2/7·14-s + 1.03·16-s − 1.39·17-s + 7/6·18-s − 3.63·19-s + 8/7·21-s − 0.785·22-s + 0.207·23-s + 9/8·24-s + 1/5·25-s + 2·27-s + 9/14·28-s − 3.42·29-s + 3.63·31-s + 0.0703·32-s − 3.14·33-s − 0.695·34-s + 21/8·36-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 5^{4} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(5-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 5^{4} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(2.216409536\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.216409536\) |
| \(L(3)\) |
|
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 | $C_2$ | \( ( 1 - p^{2} T + p^{3} T^{2} )^{2} \) |
| 5 | $C_2^2$ | \( 1 - p^{3} T^{2} + p^{6} T^{4} \) |
| 7 | $C_2^2$ | \( 1 - 4 p T - 33 p^{2} T^{2} - 4 p^{5} T^{3} + p^{8} T^{4} \) |
| good | 2 | $D_4\times C_2$ | \( 1 - p T - 7 p T^{2} + 7 p^{2} T^{3} + p^{2} T^{4} + 7 p^{6} T^{5} - 7 p^{9} T^{6} - p^{13} T^{7} + p^{16} T^{8} \) |
| 11 | $D_4\times C_2$ | \( 1 + 190 T + 2128 T^{2} + 891100 T^{3} + 411197803 T^{4} + 891100 p^{4} T^{5} + 2128 p^{8} T^{6} + 190 p^{12} T^{7} + p^{16} T^{8} \) |
| 13 | $D_4\times C_2$ | \( 1 - 80458 T^{2} + 3047580123 T^{4} - 80458 p^{8} T^{6} + p^{16} T^{8} \) |
| 17 | $D_4\times C_2$ | \( 1 + 402 T + 226772 T^{2} + 69507408 T^{3} + 27419145003 T^{4} + 69507408 p^{4} T^{5} + 226772 p^{8} T^{6} + 402 p^{12} T^{7} + p^{16} T^{8} \) |
| 19 | $D_4\times C_2$ | \( 1 + 1314 T + 962057 T^{2} + 507893850 T^{3} + 207421918356 T^{4} + 507893850 p^{4} T^{5} + 962057 p^{8} T^{6} + 1314 p^{12} T^{7} + p^{16} T^{8} \) |
| 23 | $D_4\times C_2$ | \( 1 - 110 T - 80672 T^{2} + 51360100 T^{3} - 73090417397 T^{4} + 51360100 p^{4} T^{5} - 80672 p^{8} T^{6} - 110 p^{12} T^{7} + p^{16} T^{8} \) |
| 29 | $D_{4}$ | \( ( 1 + 1442 T + 1931028 T^{2} + 1442 p^{4} T^{3} + p^{8} T^{4} )^{2} \) |
| 31 | $D_4\times C_2$ | \( 1 - 3498 T + 6685457 T^{2} - 9118547922 T^{3} + 9709193403108 T^{4} - 9118547922 p^{4} T^{5} + 6685457 p^{8} T^{6} - 3498 p^{12} T^{7} + p^{16} T^{8} \) |
| 37 | $D_4\times C_2$ | \( 1 + 2380 T + 1858993 T^{2} + 135862300 T^{3} + 352645413328 T^{4} + 135862300 p^{4} T^{5} + 1858993 p^{8} T^{6} + 2380 p^{12} T^{7} + p^{16} T^{8} \) |
| 41 | $D_4\times C_2$ | \( 1 - 4708468 T^{2} + 10678388882538 T^{4} - 4708468 p^{8} T^{6} + p^{16} T^{8} \) |
| 43 | $D_{4}$ | \( ( 1 + 56 T + 6409971 T^{2} + 56 p^{4} T^{3} + p^{8} T^{4} )^{2} \) |
| 47 | $D_4\times C_2$ | \( 1 + 8220 T + 37854542 T^{2} + 126026919240 T^{3} + 321155105319603 T^{4} + 126026919240 p^{4} T^{5} + 37854542 p^{8} T^{6} + 8220 p^{12} T^{7} + p^{16} T^{8} \) |
| 53 | $D_4\times C_2$ | \( 1 - 6548 T + 16964326 T^{2} - 66337892768 T^{3} + 270856251398899 T^{4} - 66337892768 p^{4} T^{5} + 16964326 p^{8} T^{6} - 6548 p^{12} T^{7} + p^{16} T^{8} \) |
| 59 | $D_4\times C_2$ | \( 1 + 14994 T + 117841292 T^{2} + 643261792320 T^{3} + 2601764566782411 T^{4} + 643261792320 p^{4} T^{5} + 117841292 p^{8} T^{6} + 14994 p^{12} T^{7} + p^{16} T^{8} \) |
| 61 | $D_4\times C_2$ | \( 1 - 14892 T + 120070622 T^{2} - 687217162728 T^{3} + 2961352145219283 T^{4} - 687217162728 p^{4} T^{5} + 120070622 p^{8} T^{6} - 14892 p^{12} T^{7} + p^{16} T^{8} \) |
| 67 | $D_4\times C_2$ | \( 1 - 10976 T + 56138725 T^{2} - 263770940384 T^{3} + 1305119250775864 T^{4} - 263770940384 p^{4} T^{5} + 56138725 p^{8} T^{6} - 10976 p^{12} T^{7} + p^{16} T^{8} \) |
| 71 | $D_{4}$ | \( ( 1 - 1558 T + 36698988 T^{2} - 1558 p^{4} T^{3} + p^{8} T^{4} )^{2} \) |
| 73 | $D_4\times C_2$ | \( 1 + 20472 T + 222907517 T^{2} + 1703405290008 T^{3} + 10084136982388488 T^{4} + 1703405290008 p^{4} T^{5} + 222907517 p^{8} T^{6} + 20472 p^{12} T^{7} + p^{16} T^{8} \) |
| 79 | $D_4\times C_2$ | \( 1 + 3346 T - 13976735 T^{2} - 176426921006 T^{3} - 1319530565612156 T^{4} - 176426921006 p^{4} T^{5} - 13976735 p^{8} T^{6} + 3346 p^{12} T^{7} + p^{16} T^{8} \) |
| 83 | $D_4\times C_2$ | \( 1 - 139735300 T^{2} + 8918929548919722 T^{4} - 139735300 p^{8} T^{6} + p^{16} T^{8} \) |
| 89 | $D_4\times C_2$ | \( 1 + 18258 T + 204973172 T^{2} + 1713604297872 T^{3} + 11843973346917483 T^{4} + 1713604297872 p^{4} T^{5} + 204973172 p^{8} T^{6} + 18258 p^{12} T^{7} + p^{16} T^{8} \) |
| 97 | $D_4\times C_2$ | \( 1 - 274448188 T^{2} + 34477876586097798 T^{4} - 274448188 p^{8} T^{6} + p^{16} T^{8} \) |
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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
−9.369825899304534914628175460779, −8.911640173547555292841789405596, −8.572267476622992637475636640449, −8.434705972264192198918591261061, −8.377773528830004654826685446665, −8.116061759950962271306297797335, −7.68315081525078221554754768483, −7.34698240558519189661342797157, −7.08557057449982271991967110850, −6.66157298888533757831836881464, −6.41293956904422227538911850103, −6.36297882898380412119591792836, −5.55213750179350744526136684796, −5.20160013771958094817711388834, −5.10152671412357062554814777248, −4.18521146670573224485971110344, −4.14237862245461523892096547756, −4.11866686476100255460782383589, −3.26999457470812284780708085839, −2.75474393404208831326026912025, −2.68324550292723273119663134880, −2.13624186719286563763614658423, −1.87498863376990627388934076520, −1.57489141512178176879088233372, −0.19273756153615353271697362265,
0.19273756153615353271697362265, 1.57489141512178176879088233372, 1.87498863376990627388934076520, 2.13624186719286563763614658423, 2.68324550292723273119663134880, 2.75474393404208831326026912025, 3.26999457470812284780708085839, 4.11866686476100255460782383589, 4.14237862245461523892096547756, 4.18521146670573224485971110344, 5.10152671412357062554814777248, 5.20160013771958094817711388834, 5.55213750179350744526136684796, 6.36297882898380412119591792836, 6.41293956904422227538911850103, 6.66157298888533757831836881464, 7.08557057449982271991967110850, 7.34698240558519189661342797157, 7.68315081525078221554754768483, 8.116061759950962271306297797335, 8.377773528830004654826685446665, 8.434705972264192198918591261061, 8.572267476622992637475636640449, 8.911640173547555292841789405596, 9.369825899304534914628175460779
