| L(s) = 1 | + 72·7-s − 16·13-s − 832·19-s + 1.95e3·25-s + 1.30e3·31-s + 2.20e3·37-s − 752·43-s − 6.50e3·49-s − 1.88e3·61-s − 4.43e3·67-s + 8·73-s − 1.50e4·79-s − 1.15e3·91-s − 1.14e4·97-s − 2.18e4·103-s + 6.26e4·109-s + 3.30e4·121-s + ⋯ |
| L(s) = 1 | + 1.46·7-s − 0.0946·13-s − 2.30·19-s + 3.12·25-s + 1.35·31-s + 1.61·37-s − 0.406·43-s − 2.70·49-s − 0.507·61-s − 0.987·67-s + 0.00150·73-s − 2.41·79-s − 0.139·91-s − 1.21·97-s − 2.06·103-s + 5.27·109-s + 2.25·121-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 3^{24}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(5-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 3^{24}\right)^{s/2} \, \Gamma_{\C}(s+2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(5.144241229\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.144241229\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 - 1956 T^{2} + 1749898 T^{4} - 934433424 T^{6} + 472017329619 T^{8} - 934433424 p^{8} T^{10} + 1749898 p^{16} T^{12} - 1956 p^{24} T^{14} + p^{32} T^{16} \) |
| 7 | \( ( 1 - 36 T + 106 p^{2} T^{2} - 30672 p T^{3} + 275091 p^{2} T^{4} - 30672 p^{5} T^{5} + 106 p^{10} T^{6} - 36 p^{12} T^{7} + p^{16} T^{8} )^{2} \) |
| 11 | \( 1 - 33060 T^{2} + 519732970 T^{4} - 151109656560 p T^{6} - 26166174579088653 T^{8} - 151109656560 p^{9} T^{10} + 519732970 p^{16} T^{12} - 33060 p^{24} T^{14} + p^{32} T^{16} \) |
| 13 | \( ( 1 + 8 T + 73084 T^{2} + 824888 T^{3} + 2553059014 T^{4} + 824888 p^{4} T^{5} + 73084 p^{8} T^{6} + 8 p^{12} T^{7} + p^{16} T^{8} )^{2} \) |
| 17 | \( 1 - 572856 T^{2} + 149376570268 T^{4} - 23251546985775624 T^{6} + \)\(23\!\cdots\!78\)\( T^{8} - 23251546985775624 p^{8} T^{10} + 149376570268 p^{16} T^{12} - 572856 p^{24} T^{14} + p^{32} T^{16} \) |
| 19 | \( ( 1 + 416 T + 175204 T^{2} + 24743648 T^{3} + 12857645830 T^{4} + 24743648 p^{4} T^{5} + 175204 p^{8} T^{6} + 416 p^{12} T^{7} + p^{16} T^{8} )^{2} \) |
| 23 | \( 1 - 1421064 T^{2} + 1050985398556 T^{4} - 498944323687190328 T^{6} + \)\(16\!\cdots\!94\)\( T^{8} - 498944323687190328 p^{8} T^{10} + 1050985398556 p^{16} T^{12} - 1421064 p^{24} T^{14} + p^{32} T^{16} \) |
| 29 | \( 1 - 3688760 T^{2} + 6328968515356 T^{4} - 6894727270588617608 T^{6} + \)\(55\!\cdots\!86\)\( T^{8} - 6894727270588617608 p^{8} T^{10} + 6328968515356 p^{16} T^{12} - 3688760 p^{24} T^{14} + p^{32} T^{16} \) |
| 31 | \( ( 1 - 652 T + 1482970 T^{2} + 502432976 T^{3} + 504583966387 T^{4} + 502432976 p^{4} T^{5} + 1482970 p^{8} T^{6} - 652 p^{12} T^{7} + p^{16} T^{8} )^{2} \) |
| 37 | \( ( 1 - 1104 T + 2728516 T^{2} + 47214288 p T^{3} + 64028075334 T^{4} + 47214288 p^{5} T^{5} + 2728516 p^{8} T^{6} - 1104 p^{12} T^{7} + p^{16} T^{8} )^{2} \) |
| 41 | \( 1 - 14159624 T^{2} + 103498916833564 T^{4} - \)\(49\!\cdots\!08\)\( T^{6} + \)\(16\!\cdots\!58\)\( T^{8} - \)\(49\!\cdots\!08\)\( p^{8} T^{10} + 103498916833564 p^{16} T^{12} - 14159624 p^{24} T^{14} + p^{32} T^{16} \) |
| 43 | \( ( 1 + 376 T + 2435164 T^{2} - 17346872 T^{3} + 5718362890630 T^{4} - 17346872 p^{4} T^{5} + 2435164 p^{8} T^{6} + 376 p^{12} T^{7} + p^{16} T^{8} )^{2} \) |
| 47 | \( 1 - 9951608 T^{2} + 42534569627932 T^{4} - \)\(15\!\cdots\!52\)\( T^{6} + \)\(87\!\cdots\!58\)\( T^{8} - \)\(15\!\cdots\!52\)\( p^{8} T^{10} + 42534569627932 p^{16} T^{12} - 9951608 p^{24} T^{14} + p^{32} T^{16} \) |
| 53 | \( 1 - 43175652 T^{2} + 856366462519114 T^{4} - \)\(10\!\cdots\!60\)\( T^{6} + \)\(96\!\cdots\!23\)\( T^{8} - \)\(10\!\cdots\!60\)\( p^{8} T^{10} + 856366462519114 p^{16} T^{12} - 43175652 p^{24} T^{14} + p^{32} T^{16} \) |
| 59 | \( 1 - 13955064 T^{2} + 329920173479836 T^{4} - \)\(33\!\cdots\!48\)\( T^{6} + \)\(64\!\cdots\!14\)\( T^{8} - \)\(33\!\cdots\!48\)\( p^{8} T^{10} + 329920173479836 p^{16} T^{12} - 13955064 p^{24} T^{14} + p^{32} T^{16} \) |
| 61 | \( ( 1 + 944 T + 27092644 T^{2} + 57293443856 T^{3} + 457761843365254 T^{4} + 57293443856 p^{4} T^{5} + 27092644 p^{8} T^{6} + 944 p^{12} T^{7} + p^{16} T^{8} )^{2} \) |
| 67 | \( ( 1 + 2216 T + 75666460 T^{2} + 124751160344 T^{3} + 2241054705579142 T^{4} + 124751160344 p^{4} T^{5} + 75666460 p^{8} T^{6} + 2216 p^{12} T^{7} + p^{16} T^{8} )^{2} \) |
| 71 | \( 1 - 94453176 T^{2} + 5544625357024924 T^{4} - \)\(21\!\cdots\!84\)\( T^{6} + \)\(64\!\cdots\!54\)\( T^{8} - \)\(21\!\cdots\!84\)\( p^{8} T^{10} + 5544625357024924 p^{16} T^{12} - 94453176 p^{24} T^{14} + p^{32} T^{16} \) |
| 73 | \( ( 1 - 4 T + 38831914 T^{2} + 223207748912 T^{3} + 617767212966643 T^{4} + 223207748912 p^{4} T^{5} + 38831914 p^{8} T^{6} - 4 p^{12} T^{7} + p^{16} T^{8} )^{2} \) |
| 79 | \( ( 1 + 7528 T + 152689084 T^{2} + 853733933656 T^{3} + 8853505305207814 T^{4} + 853733933656 p^{4} T^{5} + 152689084 p^{8} T^{6} + 7528 p^{12} T^{7} + p^{16} T^{8} )^{2} \) |
| 83 | \( 1 - 254485956 T^{2} + 32089697868250186 T^{4} - \)\(25\!\cdots\!52\)\( T^{6} + \)\(14\!\cdots\!51\)\( T^{8} - \)\(25\!\cdots\!52\)\( p^{8} T^{10} + 32089697868250186 p^{16} T^{12} - 254485956 p^{24} T^{14} + p^{32} T^{16} \) |
| 89 | \( 1 - 317851640 T^{2} + 49759734749167900 T^{4} - \)\(50\!\cdots\!36\)\( T^{6} + \)\(36\!\cdots\!54\)\( T^{8} - \)\(50\!\cdots\!36\)\( p^{8} T^{10} + 49759734749167900 p^{16} T^{12} - 317851640 p^{24} T^{14} + p^{32} T^{16} \) |
| 97 | \( ( 1 + 5716 T + 109302874 T^{2} + 1154541953968 T^{3} + 14270149448544355 T^{4} + 1154541953968 p^{4} T^{5} + 109302874 p^{8} T^{6} + 5716 p^{12} T^{7} + p^{16} T^{8} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{16} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−4.33021279713785311323924666872, −4.26565503838110833808200707101, −4.16572558691445253399144672814, −3.69395695222533742056118706403, −3.62485770932253852117682595844, −3.34018497977298179554483305456, −3.30347102997069225341119679166, −3.24427075345471036936918260547, −3.04874927724604807413080573686, −2.80771966379539908190716760166, −2.78845425392657257434281567414, −2.38426186968781247525536099797, −2.33337666920089151526830898006, −2.31645604404593253072569506930, −2.06045828863381754770817138820, −1.75571018681728769019034296341, −1.68360672633437887612063494425, −1.49669267797920990568911531520, −1.23452145639803508799188549934, −1.17170092554781190146848812614, −1.04473800733873160816431907734, −0.69184852329549260401589255828, −0.57268698471063045228076412567, −0.25913745291258009100396948015, −0.16831777522011208743724774047,
0.16831777522011208743724774047, 0.25913745291258009100396948015, 0.57268698471063045228076412567, 0.69184852329549260401589255828, 1.04473800733873160816431907734, 1.17170092554781190146848812614, 1.23452145639803508799188549934, 1.49669267797920990568911531520, 1.68360672633437887612063494425, 1.75571018681728769019034296341, 2.06045828863381754770817138820, 2.31645604404593253072569506930, 2.33337666920089151526830898006, 2.38426186968781247525536099797, 2.78845425392657257434281567414, 2.80771966379539908190716760166, 3.04874927724604807413080573686, 3.24427075345471036936918260547, 3.30347102997069225341119679166, 3.34018497977298179554483305456, 3.62485770932253852117682595844, 3.69395695222533742056118706403, 4.16572558691445253399144672814, 4.26565503838110833808200707101, 4.33021279713785311323924666872
Plot not available for L-functions of degree greater than 10.
