| L(s) = 1 | + 3·4-s + 4·7-s − 16·13-s + 5·16-s + 8·19-s + 12·28-s + 24·37-s + 16·43-s + 6·49-s − 48·52-s − 28·61-s − 6·64-s − 16·67-s + 48·73-s + 24·76-s + 16·79-s − 64·91-s − 12·97-s + 4·103-s + 48·109-s + 20·112-s + 30·121-s + 127-s + 131-s + 32·133-s + 137-s + 139-s + ⋯ |
| L(s) = 1 | + 3/2·4-s + 1.51·7-s − 4.43·13-s + 5/4·16-s + 1.83·19-s + 2.26·28-s + 3.94·37-s + 2.43·43-s + 6/7·49-s − 6.65·52-s − 3.58·61-s − 3/4·64-s − 1.95·67-s + 5.61·73-s + 2.75·76-s + 1.80·79-s − 6.70·91-s − 1.21·97-s + 0.394·103-s + 4.59·109-s + 1.88·112-s + 2.72·121-s + 0.0887·127-s + 0.0873·131-s + 2.77·133-s + 0.0854·137-s + 0.0848·139-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{32} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{32} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.046626505\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.046626505\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 7 | \( ( 1 - T + T^{2} )^{4} \) |
| good | 2 | \( ( 1 - 3 T^{2} + p^{2} T^{4} )^{2}( 1 + 3 T^{2} + 5 T^{4} + 3 p^{2} T^{6} + p^{4} T^{8} ) \) |
| 5 | \( ( 1 - 6 T + 18 T^{2} - 36 T^{3} + 71 T^{4} - 36 p T^{5} + 18 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} )( 1 + 6 T + 18 T^{2} + 36 T^{3} + 71 T^{4} + 36 p T^{5} + 18 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} ) \) |
| 11 | \( ( 1 - 15 T^{2} + 104 T^{4} - 15 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 13 | \( ( 1 + 4 T + 3 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{4} \) |
| 17 | \( ( 1 + 22 T^{2} + p^{2} T^{4} )^{4} \) |
| 19 | \( ( 1 - 2 T + 18 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{4} \) |
| 23 | \( ( 1 - 34 T^{2} + 627 T^{4} - 34 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 29 | \( 1 - 36 T^{2} + 634 T^{4} + 36720 T^{6} - 1358973 T^{8} + 36720 p^{2} T^{10} + 634 p^{4} T^{12} - 36 p^{6} T^{14} + p^{8} T^{16} \) |
| 31 | \( ( 1 + 22 T^{2} - 477 T^{4} + 22 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 37 | \( ( 1 - 3 T + p T^{2} )^{8} \) |
| 41 | \( 1 - 144 T^{2} + 12274 T^{4} - 734400 T^{6} + 33922467 T^{8} - 734400 p^{2} T^{10} + 12274 p^{4} T^{12} - 144 p^{6} T^{14} + p^{8} T^{16} \) |
| 43 | \( ( 1 - 8 T - 17 T^{2} + 40 T^{3} + 1960 T^{4} + 40 p T^{5} - 17 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 47 | \( 1 - 8 T^{2} + 2434 T^{4} + 54304 T^{6} + 595843 T^{8} + 54304 p^{2} T^{10} + 2434 p^{4} T^{12} - 8 p^{6} T^{14} + p^{8} T^{16} \) |
| 53 | \( ( 1 + 31 T^{2} + p^{2} T^{4} )^{4} \) |
| 59 | \( ( 1 - 106 T^{2} + 7755 T^{4} - 106 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 61 | \( ( 1 + 14 T + 46 T^{2} + 392 T^{3} + 6823 T^{4} + 392 p T^{5} + 46 p^{2} T^{6} + 14 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 67 | \( ( 1 + 8 T - 65 T^{2} - 40 T^{3} + 7864 T^{4} - 40 p T^{5} - 65 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 71 | \( ( 1 + 134 T^{2} + 11547 T^{4} + 134 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 73 | \( ( 1 - 12 T + 98 T^{2} - 12 p T^{3} + p^{2} T^{4} )^{4} \) |
| 79 | \( ( 1 - 8 T - 89 T^{2} + 40 T^{3} + 12112 T^{4} + 40 p T^{5} - 89 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 83 | \( 1 - 200 T^{2} + 16978 T^{4} - 1848800 T^{6} + 206994163 T^{8} - 1848800 p^{2} T^{10} + 16978 p^{4} T^{12} - 200 p^{6} T^{14} + p^{8} T^{16} \) |
| 89 | \( ( 1 + 276 T^{2} + 33542 T^{4} + 276 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 97 | \( ( 1 + 6 T - 146 T^{2} - 72 T^{3} + 20223 T^{4} - 72 p T^{5} - 146 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
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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
−4.74463331932134923752167567941, −4.57886167148953342151739654364, −4.56351907045522048823588612885, −4.52631282069582426941776928991, −4.30568254418428755236488012110, −3.98218007744916012908691062825, −3.89189702369814236138557497192, −3.74442070981679594391013106955, −3.38613253853079576909874984883, −3.34287676451731890301288473686, −3.04110152996875457959427534934, −3.03205127530444233551476737746, −3.02544788330664907859961032217, −2.64461979520508626035640149573, −2.61352006719502636512040276616, −2.29940597175962469450086084124, −2.16102763606980155098972098408, −2.14627847584381257175861291655, −2.02655574121094565938772890379, −2.01145159718181003928522683572, −1.33493768224810915073284609258, −1.12299168323895143833583310758, −1.11995985233995433737106232724, −0.832746723388056974957765595927, −0.26594485493364336722179564960,
0.26594485493364336722179564960, 0.832746723388056974957765595927, 1.11995985233995433737106232724, 1.12299168323895143833583310758, 1.33493768224810915073284609258, 2.01145159718181003928522683572, 2.02655574121094565938772890379, 2.14627847584381257175861291655, 2.16102763606980155098972098408, 2.29940597175962469450086084124, 2.61352006719502636512040276616, 2.64461979520508626035640149573, 3.02544788330664907859961032217, 3.03205127530444233551476737746, 3.04110152996875457959427534934, 3.34287676451731890301288473686, 3.38613253853079576909874984883, 3.74442070981679594391013106955, 3.89189702369814236138557497192, 3.98218007744916012908691062825, 4.30568254418428755236488012110, 4.52631282069582426941776928991, 4.56351907045522048823588612885, 4.57886167148953342151739654364, 4.74463331932134923752167567941
Plot not available for L-functions of degree greater than 10.
