| L(s) = 1 | + 16·4-s − 36·9-s + 114·11-s + 172·16-s + 24·19-s − 756·29-s − 186·31-s − 576·36-s − 930·41-s + 1.82e3·44-s − 196·49-s − 462·59-s − 2.70e3·61-s + 927·64-s − 3.45e3·71-s + 384·76-s − 3.25e3·79-s + 810·81-s + 1.95e3·89-s − 4.10e3·99-s − 2.70e3·101-s − 3.47e3·109-s − 1.20e4·116-s + 1.74e3·121-s − 2.97e3·124-s + 127-s + 131-s + ⋯ |
| L(s) = 1 | + 2·4-s − 4/3·9-s + 3.12·11-s + 2.68·16-s + 0.289·19-s − 4.84·29-s − 1.07·31-s − 8/3·36-s − 3.54·41-s + 6.24·44-s − 4/7·49-s − 1.01·59-s − 5.67·61-s + 1.81·64-s − 5.76·71-s + 0.579·76-s − 4.63·79-s + 10/9·81-s + 2.32·89-s − 4.16·99-s − 2.66·101-s − 3.04·109-s − 9.68·116-s + 1.31·121-s − 2.15·124-s + 0.000698·127-s + 0.000666·131-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 5^{16} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 5^{16} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.455948968\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.455948968\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( ( 1 + p^{2} T^{2} )^{4} \) |
| 5 | \( 1 \) |
| 7 | \( ( 1 + p^{2} T^{2} )^{4} \) |
| good | 2 | \( 1 - p^{4} T^{2} + 21 p^{2} T^{4} + 481 T^{6} - 2539 p^{2} T^{8} + 481 p^{6} T^{10} + 21 p^{14} T^{12} - p^{22} T^{14} + p^{24} T^{16} \) |
| 11 | \( ( 1 - 57 T + 4001 T^{2} - 182772 T^{3} + 7206874 T^{4} - 182772 p^{3} T^{5} + 4001 p^{6} T^{6} - 57 p^{9} T^{7} + p^{12} T^{8} )^{2} \) |
| 13 | \( 1 - 7079 T^{2} + 27317214 T^{4} - 5408074117 p T^{6} + 159254918186594 T^{8} - 5408074117 p^{7} T^{10} + 27317214 p^{12} T^{12} - 7079 p^{18} T^{14} + p^{24} T^{16} \) |
| 17 | \( 1 - 17779 T^{2} + 11067450 p T^{4} - 1407976054013 T^{6} + 7815459662716778 T^{8} - 1407976054013 p^{6} T^{10} + 11067450 p^{13} T^{12} - 17779 p^{18} T^{14} + p^{24} T^{16} \) |
| 19 | \( ( 1 - 12 T - 10212 T^{2} - 154020 T^{3} + 95206646 T^{4} - 154020 p^{3} T^{5} - 10212 p^{6} T^{6} - 12 p^{9} T^{7} + p^{12} T^{8} )^{2} \) |
| 23 | \( 1 - 31828 T^{2} + 620170842 T^{4} - 8468979898736 T^{6} + 101809268764343459 T^{8} - 8468979898736 p^{6} T^{10} + 620170842 p^{12} T^{12} - 31828 p^{18} T^{14} + p^{24} T^{16} \) |
| 29 | \( ( 1 + 378 T + 143760 T^{2} + 1025172 p T^{3} + 5852097929 T^{4} + 1025172 p^{4} T^{5} + 143760 p^{6} T^{6} + 378 p^{9} T^{7} + p^{12} T^{8} )^{2} \) |
| 31 | \( ( 1 + 3 p T + 16330 T^{2} - 936891 T^{3} + 385099458 T^{4} - 936891 p^{3} T^{5} + 16330 p^{6} T^{6} + 3 p^{10} T^{7} + p^{12} T^{8} )^{2} \) |
| 37 | \( 1 - 207809 T^{2} + 23288320585 T^{4} - 1818846583640678 T^{6} + \)\(10\!\cdots\!58\)\( T^{8} - 1818846583640678 p^{6} T^{10} + 23288320585 p^{12} T^{12} - 207809 p^{18} T^{14} + p^{24} T^{16} \) |
| 41 | \( ( 1 + 465 T + 105716 T^{2} + 566235 T^{3} - 2050285754 T^{4} + 566235 p^{3} T^{5} + 105716 p^{6} T^{6} + 465 p^{9} T^{7} + p^{12} T^{8} )^{2} \) |
| 43 | \( 1 - 125228 T^{2} + 15455427282 T^{4} - 1545044296825216 T^{6} + \)\(15\!\cdots\!19\)\( T^{8} - 1545044296825216 p^{6} T^{10} + 15455427282 p^{12} T^{12} - 125228 p^{18} T^{14} + p^{24} T^{16} \) |
| 47 | \( 1 - 623016 T^{2} + 180431963644 T^{4} - 32308674389768664 T^{6} + \)\(39\!\cdots\!54\)\( T^{8} - 32308674389768664 p^{6} T^{10} + 180431963644 p^{12} T^{12} - 623016 p^{18} T^{14} + p^{24} T^{16} \) |
| 53 | \( 1 - 135451 T^{2} + 38529689730 T^{4} - 8651841044933597 T^{6} + \)\(85\!\cdots\!18\)\( T^{8} - 8651841044933597 p^{6} T^{10} + 38529689730 p^{12} T^{12} - 135451 p^{18} T^{14} + p^{24} T^{16} \) |
| 59 | \( ( 1 + 231 T + 701280 T^{2} + 148812447 T^{3} + 204019205798 T^{4} + 148812447 p^{3} T^{5} + 701280 p^{6} T^{6} + 231 p^{9} T^{7} + p^{12} T^{8} )^{2} \) |
| 61 | \( ( 1 + 1353 T + 1374808 T^{2} + 931878711 T^{3} + 511309944510 T^{4} + 931878711 p^{3} T^{5} + 1374808 p^{6} T^{6} + 1353 p^{9} T^{7} + p^{12} T^{8} )^{2} \) |
| 67 | \( 1 - 995573 T^{2} + 706524334905 T^{4} - 318789658849139158 T^{6} + \)\(11\!\cdots\!14\)\( T^{8} - 318789658849139158 p^{6} T^{10} + 706524334905 p^{12} T^{12} - 995573 p^{18} T^{14} + p^{24} T^{16} \) |
| 71 | \( ( 1 + 1725 T + 1434819 T^{2} + 706812600 T^{3} + 347044577276 T^{4} + 706812600 p^{3} T^{5} + 1434819 p^{6} T^{6} + 1725 p^{9} T^{7} + p^{12} T^{8} )^{2} \) |
| 73 | \( 1 - 1799816 T^{2} + 1700330801820 T^{4} - 1053085527973751032 T^{6} + \)\(47\!\cdots\!58\)\( T^{8} - 1053085527973751032 p^{6} T^{10} + 1700330801820 p^{12} T^{12} - 1799816 p^{18} T^{14} + p^{24} T^{16} \) |
| 79 | \( ( 1 + 1629 T + 1989837 T^{2} + 1837769472 T^{3} + 1504809477470 T^{4} + 1837769472 p^{3} T^{5} + 1989837 p^{6} T^{6} + 1629 p^{9} T^{7} + p^{12} T^{8} )^{2} \) |
| 83 | \( 1 - 1219827 T^{2} + 504702020110 T^{4} - 464473092130980597 T^{6} + \)\(44\!\cdots\!74\)\( T^{8} - 464473092130980597 p^{6} T^{10} + 504702020110 p^{12} T^{12} - 1219827 p^{18} T^{14} + p^{24} T^{16} \) |
| 89 | \( ( 1 - 978 T + 526228 T^{2} + 159100050 T^{3} - 360762420714 T^{4} + 159100050 p^{3} T^{5} + 526228 p^{6} T^{6} - 978 p^{9} T^{7} + p^{12} T^{8} )^{2} \) |
| 97 | \( 1 - 3522744 T^{2} + 6104761341340 T^{4} - 7977627762356281608 T^{6} + \)\(83\!\cdots\!78\)\( T^{8} - 7977627762356281608 p^{6} T^{10} + 6104761341340 p^{12} T^{12} - 3522744 p^{18} T^{14} + p^{24} T^{16} \) |
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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
−4.24642887871162799198089156343, −4.06087226550103561327642990840, −4.02708928310756502997569174258, −3.82349417864404875823871011415, −3.70658728036858214765437819195, −3.60497443156334861427989107161, −3.11976373241817215720208676246, −3.10545311818635751354621455551, −3.10256828086724367134345243643, −3.09256068706798188299707909614, −2.90745631416663258478469941981, −2.76136498128472708668165739882, −2.75015863114638450456134015097, −1.96365098473575479591999755737, −1.90564771711877786173905054298, −1.79744278634277509679611888637, −1.71330395496299361581308158337, −1.66086047957781934550068005068, −1.60531970990812225045947878584, −1.53496738894292689205369990934, −1.21305017290905210097786579535, −0.849552193957530345526195270976, −0.36786934100420709042646277334, −0.35446008319296379254723450420, −0.12598167964443848909236914741,
0.12598167964443848909236914741, 0.35446008319296379254723450420, 0.36786934100420709042646277334, 0.849552193957530345526195270976, 1.21305017290905210097786579535, 1.53496738894292689205369990934, 1.60531970990812225045947878584, 1.66086047957781934550068005068, 1.71330395496299361581308158337, 1.79744278634277509679611888637, 1.90564771711877786173905054298, 1.96365098473575479591999755737, 2.75015863114638450456134015097, 2.76136498128472708668165739882, 2.90745631416663258478469941981, 3.09256068706798188299707909614, 3.10256828086724367134345243643, 3.10545311818635751354621455551, 3.11976373241817215720208676246, 3.60497443156334861427989107161, 3.70658728036858214765437819195, 3.82349417864404875823871011415, 4.02708928310756502997569174258, 4.06087226550103561327642990840, 4.24642887871162799198089156343
Plot not available for L-functions of degree greater than 10.
