| L(s) = 1 | + 10·4-s + 36·13-s + 71·16-s + 12·19-s + 46·25-s − 28·31-s − 4·37-s − 152·43-s + 28·49-s + 360·52-s + 180·61-s + 348·64-s − 132·67-s + 272·73-s + 120·76-s + 316·79-s − 364·97-s + 460·100-s − 296·103-s − 316·109-s + 352·121-s − 280·124-s + 127-s + 131-s + 137-s + 139-s − 40·148-s + ⋯ |
| L(s) = 1 | + 5/2·4-s + 2.76·13-s + 4.43·16-s + 0.631·19-s + 1.83·25-s − 0.903·31-s − 0.108·37-s − 3.53·43-s + 4/7·49-s + 6.92·52-s + 2.95·61-s + 5.43·64-s − 1.97·67-s + 3.72·73-s + 1.57·76-s + 4·79-s − 3.75·97-s + 23/5·100-s − 2.87·103-s − 2.89·109-s + 2.90·121-s − 2.25·124-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-s + 0.00719·139-s − 0.270·148-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{24} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{24} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+1)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(17.02124434\) |
| \(L(\frac12)\) |
\(\approx\) |
\(17.02124434\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 7 | \( ( 1 - p T^{2} )^{4} \) |
| good | 2 | \( ( 1 - p^{2} T + 3 T^{2} + 7 p T^{3} - 23 p T^{4} + 7 p^{3} T^{5} + 3 p^{4} T^{6} - p^{8} T^{7} + p^{8} T^{8} )( 1 + p^{2} T + 3 T^{2} - 7 p T^{3} - 23 p T^{4} - 7 p^{3} T^{5} + 3 p^{4} T^{6} + p^{8} T^{7} + p^{8} T^{8} ) \) |
| 5 | \( 1 - 46 T^{2} + 517 p T^{4} - 15006 p T^{6} + 2448884 T^{8} - 15006 p^{5} T^{10} + 517 p^{9} T^{12} - 46 p^{12} T^{14} + p^{16} T^{16} \) |
| 11 | \( 1 - 32 p T^{2} + 60608 T^{4} - 5817744 T^{6} + 536124638 T^{8} - 5817744 p^{4} T^{10} + 60608 p^{8} T^{12} - 32 p^{13} T^{14} + p^{16} T^{16} \) |
| 13 | \( ( 1 - 18 T + 203 T^{2} - 1116 T^{3} - 15162 T^{4} - 1116 p^{2} T^{5} + 203 p^{4} T^{6} - 18 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
| 17 | \( 1 - 1436 T^{2} + 923698 T^{4} - 370294112 T^{6} + 115162781275 T^{8} - 370294112 p^{4} T^{10} + 923698 p^{8} T^{12} - 1436 p^{12} T^{14} + p^{16} T^{16} \) |
| 19 | \( ( 1 - 6 T + 686 T^{2} - 9930 T^{3} + 14574 p T^{4} - 9930 p^{2} T^{5} + 686 p^{4} T^{6} - 6 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
| 23 | \( 1 - 2618 T^{2} + 3358561 T^{4} - 2784726434 T^{6} + 1689891029188 T^{8} - 2784726434 p^{4} T^{10} + 3358561 p^{8} T^{12} - 2618 p^{12} T^{14} + p^{16} T^{16} \) |
| 29 | \( 1 - 4618 T^{2} + 10050413 T^{4} - 16426050 p^{2} T^{6} + 13515667453952 T^{8} - 16426050 p^{6} T^{10} + 10050413 p^{8} T^{12} - 4618 p^{12} T^{14} + p^{16} T^{16} \) |
| 31 | \( ( 1 + 14 T + 2591 T^{2} + 48888 T^{3} + 3165494 T^{4} + 48888 p^{2} T^{5} + 2591 p^{4} T^{6} + 14 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
| 37 | \( ( 1 + 2 T + 2585 T^{2} + 34806 T^{3} + 4143440 T^{4} + 34806 p^{2} T^{5} + 2585 p^{4} T^{6} + 2 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
| 41 | \( 1 - 6358 T^{2} + 22869017 T^{4} - 54858246342 T^{6} + 103673363890388 T^{8} - 54858246342 p^{4} T^{10} + 22869017 p^{8} T^{12} - 6358 p^{12} T^{14} + p^{16} T^{16} \) |
| 43 | \( ( 1 + 76 T + 5314 T^{2} + 190816 T^{3} + 9873739 T^{4} + 190816 p^{2} T^{5} + 5314 p^{4} T^{6} + 76 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
| 47 | \( 1 - 8246 T^{2} + 35831041 T^{4} - 113409108974 T^{6} + 281505221123332 T^{8} - 113409108974 p^{4} T^{10} + 35831041 p^{8} T^{12} - 8246 p^{12} T^{14} + p^{16} T^{16} \) |
| 53 | \( 1 - 10274 T^{2} + 52217209 T^{4} - 208954724186 T^{6} + 680836729259764 T^{8} - 208954724186 p^{4} T^{10} + 52217209 p^{8} T^{12} - 10274 p^{12} T^{14} + p^{16} T^{16} \) |
| 59 | \( 1 - 9908 T^{2} + 40995418 T^{4} - 41869627760 T^{6} - 91311998425373 T^{8} - 41869627760 p^{4} T^{10} + 40995418 p^{8} T^{12} - 9908 p^{12} T^{14} + p^{16} T^{16} \) |
| 61 | \( ( 1 - 90 T + 17030 T^{2} - 997326 T^{3} + 98518794 T^{4} - 997326 p^{2} T^{5} + 17030 p^{4} T^{6} - 90 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
| 67 | \( ( 1 + 66 T + 6185 T^{2} + 321306 T^{3} + 39824424 T^{4} + 321306 p^{2} T^{5} + 6185 p^{4} T^{6} + 66 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
| 71 | \( 1 - 25586 T^{2} + 289157185 T^{4} - 2023738793714 T^{6} + 10971294311394052 T^{8} - 2023738793714 p^{4} T^{10} + 289157185 p^{8} T^{12} - 25586 p^{12} T^{14} + p^{16} T^{16} \) |
| 73 | \( ( 1 - 136 T + 18272 T^{2} - 1793928 T^{3} + 146961182 T^{4} - 1793928 p^{2} T^{5} + 18272 p^{4} T^{6} - 136 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
| 79 | \( ( 1 - 2 p T + 31657 T^{2} - 2953874 T^{3} + 313797844 T^{4} - 2953874 p^{2} T^{5} + 31657 p^{4} T^{6} - 2 p^{7} T^{7} + p^{8} T^{8} )^{2} \) |
| 83 | \( 1 - 32054 T^{2} + 519953785 T^{4} - 5607497327366 T^{6} + 44492093491370932 T^{8} - 5607497327366 p^{4} T^{10} + 519953785 p^{8} T^{12} - 32054 p^{12} T^{14} + p^{16} T^{16} \) |
| 89 | \( 1 - 44350 T^{2} + 947747465 T^{4} - 12796017692742 T^{6} + 120000800313605876 T^{8} - 12796017692742 p^{4} T^{10} + 947747465 p^{8} T^{12} - 44350 p^{12} T^{14} + p^{16} T^{16} \) |
| 97 | \( ( 1 + 182 T + 34250 T^{2} + 3401706 T^{3} + 407518010 T^{4} + 3401706 p^{2} T^{5} + 34250 p^{4} T^{6} + 182 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
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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
−5.42540274591392708659735991127, −5.37546610529332607986084618053, −5.35700457484492641854227630263, −5.25385544539426153523793331483, −4.81493181686830312357725408420, −4.62510932171510529676916049330, −4.56242662922455167796602471835, −4.22695234719187201905166543278, −4.19192259538390413989128578290, −3.68475919885661763436679225156, −3.57122058266748149423559425346, −3.54329925873822736092938473837, −3.42124542233371734198814536990, −3.37459480461844376771359875881, −3.03752195309893325645102117568, −2.82224312678982177941555818626, −2.63859987119320154898442511320, −2.28398247401884222969156153547, −2.14664810008875027767639423945, −1.88366755440864623884734840587, −1.73358107548731186783109328337, −1.23281831665402012666621442971, −1.12814751984459438724577775041, −1.08255699745915902361352106185, −0.46016566572886150955268687412,
0.46016566572886150955268687412, 1.08255699745915902361352106185, 1.12814751984459438724577775041, 1.23281831665402012666621442971, 1.73358107548731186783109328337, 1.88366755440864623884734840587, 2.14664810008875027767639423945, 2.28398247401884222969156153547, 2.63859987119320154898442511320, 2.82224312678982177941555818626, 3.03752195309893325645102117568, 3.37459480461844376771359875881, 3.42124542233371734198814536990, 3.54329925873822736092938473837, 3.57122058266748149423559425346, 3.68475919885661763436679225156, 4.19192259538390413989128578290, 4.22695234719187201905166543278, 4.56242662922455167796602471835, 4.62510932171510529676916049330, 4.81493181686830312357725408420, 5.25385544539426153523793331483, 5.35700457484492641854227630263, 5.37546610529332607986084618053, 5.42540274591392708659735991127
Plot not available for L-functions of degree greater than 10.
