| L(s) = 1 | + 2·3-s + 4-s + 6·5-s + 9-s + 2·12-s + 12·15-s + 4·16-s + 6·20-s − 48·23-s + 19·25-s − 8·31-s + 36-s + 22·37-s + 6·45-s + 8·48-s + 2·49-s + 18·53-s + 12·59-s + 12·60-s − 16·67-s − 96·69-s + 12·71-s + 38·75-s + 24·80-s + 72·89-s − 48·92-s − 16·93-s + ⋯ |
| L(s) = 1 | + 1.15·3-s + 1/2·4-s + 2.68·5-s + 1/3·9-s + 0.577·12-s + 3.09·15-s + 16-s + 1.34·20-s − 10.0·23-s + 19/5·25-s − 1.43·31-s + 1/6·36-s + 3.61·37-s + 0.894·45-s + 1.15·48-s + 2/7·49-s + 2.47·53-s + 1.56·59-s + 1.54·60-s − 1.95·67-s − 11.5·69-s + 1.42·71-s + 4.38·75-s + 2.68·80-s + 7.63·89-s − 5.00·92-s − 1.65·93-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 11^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 11^{16}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(9.238487029\) |
| \(L(\frac12)\) |
\(\approx\) |
\(9.238487029\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2} \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 - T^{2} - 3 T^{4} + 7 T^{6} + 5 T^{8} + 7 p^{2} T^{10} - 3 p^{4} T^{12} - p^{6} T^{14} + p^{8} T^{16} \) |
| 5 | \( ( 1 - 3 T + 4 T^{2} + 3 T^{3} - 29 T^{4} + 3 p T^{5} + 4 p^{2} T^{6} - 3 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 7 | \( 1 - 2 T^{2} - 45 T^{4} + 188 T^{6} + 1829 T^{8} + 188 p^{2} T^{10} - 45 p^{4} T^{12} - 2 p^{6} T^{14} + p^{8} T^{16} \) |
| 13 | \( 1 - 23 T^{2} + 360 T^{4} - 4393 T^{6} + 40199 T^{8} - 4393 p^{2} T^{10} + 360 p^{4} T^{12} - 23 p^{6} T^{14} + p^{8} T^{16} \) |
| 17 | \( 1 - 31 T^{2} + 672 T^{4} - 11873 T^{6} + 173855 T^{8} - 11873 p^{2} T^{10} + 672 p^{4} T^{12} - 31 p^{6} T^{14} + p^{8} T^{16} \) |
| 19 | \( 1 + 10 T^{2} - 261 T^{4} - 6220 T^{6} + 32021 T^{8} - 6220 p^{2} T^{10} - 261 p^{4} T^{12} + 10 p^{6} T^{14} + p^{8} T^{16} \) |
| 23 | \( ( 1 + 6 T + p T^{2} )^{8} \) |
| 29 | \( 1 - 55 T^{2} + 2184 T^{4} - 73865 T^{6} + 2225831 T^{8} - 73865 p^{2} T^{10} + 2184 p^{4} T^{12} - 55 p^{6} T^{14} + p^{8} T^{16} \) |
| 31 | \( ( 1 + 4 T - 15 T^{2} - 184 T^{3} - 271 T^{4} - 184 p T^{5} - 15 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 37 | \( ( 1 - 11 T + 84 T^{2} - 517 T^{3} + 2579 T^{4} - 517 p T^{5} + 84 p^{2} T^{6} - 11 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 41 | \( 1 - 79 T^{2} + 4560 T^{4} - 227441 T^{6} + 10302479 T^{8} - 227441 p^{2} T^{10} + 4560 p^{4} T^{12} - 79 p^{6} T^{14} + p^{8} T^{16} \) |
| 43 | \( ( 1 + 74 T^{2} + p^{2} T^{4} )^{4} \) |
| 47 | \( ( 1 - p T^{2} + p^{2} T^{4} - p^{3} T^{6} + p^{4} T^{8} )^{2} \) |
| 53 | \( ( 1 - 9 T + 28 T^{2} + 225 T^{3} - 3509 T^{4} + 225 p T^{5} + 28 p^{2} T^{6} - 9 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 59 | \( ( 1 - 6 T - 23 T^{2} + 492 T^{3} - 1595 T^{4} + 492 p T^{5} - 23 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 61 | \( ( 1 - p T^{2} + p^{2} T^{4} - p^{3} T^{6} + p^{4} T^{8} )^{2} \) |
| 67 | \( ( 1 + 2 T + p T^{2} )^{8} \) |
| 71 | \( ( 1 - 6 T - 35 T^{2} + 636 T^{3} - 1331 T^{4} + 636 p T^{5} - 35 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 73 | \( 1 - 98 T^{2} + 4275 T^{4} + 103292 T^{6} - 32904091 T^{8} + 103292 p^{2} T^{10} + 4275 p^{4} T^{12} - 98 p^{6} T^{14} + p^{8} T^{16} \) |
| 79 | \( ( 1 - p T^{2} + p^{2} T^{4} - p^{3} T^{6} + p^{4} T^{8} )^{2} \) |
| 83 | \( ( 1 - p T^{2} + p^{2} T^{4} - p^{3} T^{6} + p^{4} T^{8} )^{2} \) |
| 89 | \( ( 1 - 9 T + p T^{2} )^{8} \) |
| 97 | \( ( 1 - 7 T - 48 T^{2} + 1015 T^{3} - 2449 T^{4} + 1015 p T^{5} - 48 p^{2} T^{6} - 7 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
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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
−5.13477699623572997110190246485, −5.08357657613375396155039659174, −4.80527792630616643529422370757, −4.64389207856629246177084570818, −4.44561055528657709223098012771, −4.15493862624674070370061949287, −4.05061497994884935023729049188, −3.97713371202421931921103275932, −3.88575524645190048200182066540, −3.84465849175569065236856702341, −3.76755041698659470355850878710, −3.26128540389374304852470714138, −3.19077166505048681194876320322, −3.04132081423641568611324830354, −2.80947089584478845843951915826, −2.40444172513942350548236196685, −2.35394230324454250808928624287, −2.15701784275697087471445720558, −2.14891460825547206488438341386, −2.00993323606650497946616054262, −1.88294261858518049186058766262, −1.74097207199372217932470724467, −1.43492699702706317149551454200, −0.69066536999235802947591352818, −0.61468849746258583656114674166,
0.61468849746258583656114674166, 0.69066536999235802947591352818, 1.43492699702706317149551454200, 1.74097207199372217932470724467, 1.88294261858518049186058766262, 2.00993323606650497946616054262, 2.14891460825547206488438341386, 2.15701784275697087471445720558, 2.35394230324454250808928624287, 2.40444172513942350548236196685, 2.80947089584478845843951915826, 3.04132081423641568611324830354, 3.19077166505048681194876320322, 3.26128540389374304852470714138, 3.76755041698659470355850878710, 3.84465849175569065236856702341, 3.88575524645190048200182066540, 3.97713371202421931921103275932, 4.05061497994884935023729049188, 4.15493862624674070370061949287, 4.44561055528657709223098012771, 4.64389207856629246177084570818, 4.80527792630616643529422370757, 5.08357657613375396155039659174, 5.13477699623572997110190246485
Plot not available for L-functions of degree greater than 10.
