| L(s) = 1 | + 4·3-s − 4·5-s + 6·7-s + 6·9-s − 13-s − 16·15-s − 7·17-s + 7·19-s + 24·21-s + 10·23-s + 6·25-s − 20·29-s + 7·31-s − 24·35-s − 2·37-s − 4·39-s − 3·41-s − 4·43-s − 24·45-s − 14·47-s + 35·49-s − 28·51-s + 11·53-s + 28·57-s + 6·59-s + 12·61-s + 36·63-s + ⋯ |
| L(s) = 1 | + 2.30·3-s − 1.78·5-s + 2.26·7-s + 2·9-s − 0.277·13-s − 4.13·15-s − 1.69·17-s + 1.60·19-s + 5.23·21-s + 2.08·23-s + 6/5·25-s − 3.71·29-s + 1.25·31-s − 4.05·35-s − 0.328·37-s − 0.640·39-s − 0.468·41-s − 0.609·43-s − 3.57·45-s − 2.04·47-s + 5·49-s − 3.92·51-s + 1.51·53-s + 3.70·57-s + 0.781·59-s + 1.53·61-s + 4.53·63-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{8} \cdot 5^{8} \cdot 31^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{8} \cdot 5^{8} \cdot 31^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(16.63644752\) |
| \(L(\frac12)\) |
\(\approx\) |
\(16.63644752\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( ( 1 - T + T^{2} )^{4} \) |
| 5 | \( ( 1 + T + T^{2} )^{4} \) |
| 31 | \( 1 - 7 T + 67 T^{2} - 479 T^{3} + 2399 T^{4} - 479 p T^{5} + 67 p^{2} T^{6} - 7 p^{3} T^{7} + p^{4} T^{8} \) |
| good | 7 | \( 1 - 6 T + T^{2} + 4 p T^{3} + 125 T^{4} - 366 T^{5} - 131 p T^{6} + 4 p T^{7} + 12139 T^{8} + 4 p^{2} T^{9} - 131 p^{3} T^{10} - 366 p^{3} T^{11} + 125 p^{4} T^{12} + 4 p^{6} T^{13} + p^{6} T^{14} - 6 p^{7} T^{15} + p^{8} T^{16} \) |
| 11 | \( 1 - 18 T^{2} + 16 T^{3} + 89 T^{4} - 232 T^{5} + 190 T^{6} + 1288 T^{7} + 676 T^{8} + 1288 p T^{9} + 190 p^{2} T^{10} - 232 p^{3} T^{11} + 89 p^{4} T^{12} + 16 p^{5} T^{13} - 18 p^{6} T^{14} + p^{8} T^{16} \) |
| 13 | \( 1 + T - 21 T^{2} - 54 T^{3} + 210 T^{4} + 66 p T^{5} + 3194 T^{6} - 6853 T^{7} - 69039 T^{8} - 6853 p T^{9} + 3194 p^{2} T^{10} + 66 p^{4} T^{11} + 210 p^{4} T^{12} - 54 p^{5} T^{13} - 21 p^{6} T^{14} + p^{7} T^{15} + p^{8} T^{16} \) |
| 17 | \( 1 + 7 T - 13 T^{2} - 42 T^{3} + 834 T^{4} - 14 T^{5} - 19170 T^{6} - 1295 p T^{7} + 130425 T^{8} - 1295 p^{2} T^{9} - 19170 p^{2} T^{10} - 14 p^{3} T^{11} + 834 p^{4} T^{12} - 42 p^{5} T^{13} - 13 p^{6} T^{14} + 7 p^{7} T^{15} + p^{8} T^{16} \) |
| 19 | \( 1 - 7 T - 15 T^{2} + 8 p T^{3} + 394 T^{4} - 2472 T^{5} - 6220 T^{6} + 39259 T^{7} - 80391 T^{8} + 39259 p T^{9} - 6220 p^{2} T^{10} - 2472 p^{3} T^{11} + 394 p^{4} T^{12} + 8 p^{6} T^{13} - 15 p^{6} T^{14} - 7 p^{7} T^{15} + p^{8} T^{16} \) |
| 23 | \( ( 1 - 5 T + 32 T^{2} - 44 T^{3} + 558 T^{4} - 44 p T^{5} + 32 p^{2} T^{6} - 5 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 29 | \( ( 1 + 10 T + 109 T^{2} + 709 T^{3} + 4743 T^{4} + 709 p T^{5} + 109 p^{2} T^{6} + 10 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 37 | \( 1 + 2 T - 139 T^{2} - 4 p T^{3} + 12167 T^{4} + 7668 T^{5} - 699699 T^{6} - 100438 T^{7} + 30385193 T^{8} - 100438 p T^{9} - 699699 p^{2} T^{10} + 7668 p^{3} T^{11} + 12167 p^{4} T^{12} - 4 p^{6} T^{13} - 139 p^{6} T^{14} + 2 p^{7} T^{15} + p^{8} T^{16} \) |
| 41 | \( 1 + 3 T - 99 T^{2} - 678 T^{3} + 4739 T^{4} + 42117 T^{5} - 33102 T^{6} - 969027 T^{7} - 2085324 T^{8} - 969027 p T^{9} - 33102 p^{2} T^{10} + 42117 p^{3} T^{11} + 4739 p^{4} T^{12} - 678 p^{5} T^{13} - 99 p^{6} T^{14} + 3 p^{7} T^{15} + p^{8} T^{16} \) |
| 43 | \( 1 + 4 T - 91 T^{2} - 166 T^{3} + 4161 T^{4} - 4250 T^{5} - 219889 T^{6} + 188106 T^{7} + 11373235 T^{8} + 188106 p T^{9} - 219889 p^{2} T^{10} - 4250 p^{3} T^{11} + 4161 p^{4} T^{12} - 166 p^{5} T^{13} - 91 p^{6} T^{14} + 4 p^{7} T^{15} + p^{8} T^{16} \) |
| 47 | \( ( 1 + 7 T + 178 T^{2} + 906 T^{3} + 12296 T^{4} + 906 p T^{5} + 178 p^{2} T^{6} + 7 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 53 | \( 1 - 11 T + 37 T^{2} + 824 T^{3} - 11956 T^{4} + 87018 T^{5} - 63186 T^{6} - 4664919 T^{7} + 56154425 T^{8} - 4664919 p T^{9} - 63186 p^{2} T^{10} + 87018 p^{3} T^{11} - 11956 p^{4} T^{12} + 824 p^{5} T^{13} + 37 p^{6} T^{14} - 11 p^{7} T^{15} + p^{8} T^{16} \) |
| 59 | \( 1 - 6 T - 69 T^{2} + 1224 T^{3} - 2152 T^{4} - 60648 T^{5} + 462606 T^{6} + 1001121 T^{7} - 28420284 T^{8} + 1001121 p T^{9} + 462606 p^{2} T^{10} - 60648 p^{3} T^{11} - 2152 p^{4} T^{12} + 1224 p^{5} T^{13} - 69 p^{6} T^{14} - 6 p^{7} T^{15} + p^{8} T^{16} \) |
| 61 | \( ( 1 - 6 T + 81 T^{2} + 141 T^{3} + 2017 T^{4} + 141 p T^{5} + 81 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 67 | \( 1 - 225 T^{2} + 246 T^{3} + 29525 T^{4} - 35916 T^{5} - 2712321 T^{6} + 1113150 T^{7} + 196789611 T^{8} + 1113150 p T^{9} - 2712321 p^{2} T^{10} - 35916 p^{3} T^{11} + 29525 p^{4} T^{12} + 246 p^{5} T^{13} - 225 p^{6} T^{14} + p^{8} T^{16} \) |
| 71 | \( 1 + 28 T + 305 T^{2} + 2006 T^{3} + 15628 T^{4} + 105426 T^{5} + 88132 T^{6} - 1577567 T^{7} - 89512 T^{8} - 1577567 p T^{9} + 88132 p^{2} T^{10} + 105426 p^{3} T^{11} + 15628 p^{4} T^{12} + 2006 p^{5} T^{13} + 305 p^{6} T^{14} + 28 p^{7} T^{15} + p^{8} T^{16} \) |
| 73 | \( 1 + 13 T + 98 T^{2} + 2519 T^{3} + 26391 T^{4} + 170464 T^{5} + 2930120 T^{6} + 27006804 T^{7} + 150211900 T^{8} + 27006804 p T^{9} + 2930120 p^{2} T^{10} + 170464 p^{3} T^{11} + 26391 p^{4} T^{12} + 2519 p^{5} T^{13} + 98 p^{6} T^{14} + 13 p^{7} T^{15} + p^{8} T^{16} \) |
| 79 | \( 1 + 2 T - 121 T^{2} + 1504 T^{3} + 9188 T^{4} - 158672 T^{5} + 1091070 T^{6} + 10107321 T^{7} - 104427104 T^{8} + 10107321 p T^{9} + 1091070 p^{2} T^{10} - 158672 p^{3} T^{11} + 9188 p^{4} T^{12} + 1504 p^{5} T^{13} - 121 p^{6} T^{14} + 2 p^{7} T^{15} + p^{8} T^{16} \) |
| 83 | \( 1 + T - 183 T^{2} - 340 T^{3} + 14274 T^{4} + 31314 T^{5} - 972830 T^{6} - 1101325 T^{7} + 88016217 T^{8} - 1101325 p T^{9} - 972830 p^{2} T^{10} + 31314 p^{3} T^{11} + 14274 p^{4} T^{12} - 340 p^{5} T^{13} - 183 p^{6} T^{14} + p^{7} T^{15} + p^{8} T^{16} \) |
| 89 | \( ( 1 - 10 T + 71 T^{2} + 1281 T^{3} - 13275 T^{4} + 1281 p T^{5} + 71 p^{2} T^{6} - 10 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 97 | \( ( 1 - 7 T + 319 T^{2} - 1614 T^{3} + 42744 T^{4} - 1614 p T^{5} + 319 p^{2} T^{6} - 7 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
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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
−3.87241048237508530005054340268, −3.83907902207692681530262088090, −3.56172806795947812498006112303, −3.50127211064877830272822943334, −3.47757831788744457215095503074, −3.41062940417456177756060998334, −2.99894734516263879892210494603, −2.96191569398017682572590527076, −2.82333348349066873967095026461, −2.77363253219300097283075040193, −2.74188224781168589327233427351, −2.54055350111536547398424441917, −2.34975057083955503318037317146, −2.21600769280899753807831962507, −2.15242416290783036185774122517, −1.84419950380032519144263412932, −1.68933925057077363732023412066, −1.61041966553173728845021799974, −1.55098464215452802327668712984, −1.45014710598734079275472271681, −1.25210136222275578775916006238, −0.65813369371833815954837895566, −0.63673578120553434146136664896, −0.59954342527224063840189356796, −0.28269178999237350822449855495,
0.28269178999237350822449855495, 0.59954342527224063840189356796, 0.63673578120553434146136664896, 0.65813369371833815954837895566, 1.25210136222275578775916006238, 1.45014710598734079275472271681, 1.55098464215452802327668712984, 1.61041966553173728845021799974, 1.68933925057077363732023412066, 1.84419950380032519144263412932, 2.15242416290783036185774122517, 2.21600769280899753807831962507, 2.34975057083955503318037317146, 2.54055350111536547398424441917, 2.74188224781168589327233427351, 2.77363253219300097283075040193, 2.82333348349066873967095026461, 2.96191569398017682572590527076, 2.99894734516263879892210494603, 3.41062940417456177756060998334, 3.47757831788744457215095503074, 3.50127211064877830272822943334, 3.56172806795947812498006112303, 3.83907902207692681530262088090, 3.87241048237508530005054340268
Plot not available for L-functions of degree greater than 10.
