| L(s) = 1 | − 2·5-s − 3·9-s + 4·11-s − 4·19-s + 25-s + 12·29-s + 28·31-s − 20·41-s + 6·45-s − 3·49-s − 8·55-s − 16·59-s + 4·61-s + 36·71-s + 8·79-s + 6·81-s + 12·89-s + 8·95-s − 12·99-s − 4·101-s − 44·109-s + 18·121-s − 8·125-s + 127-s + 131-s + 137-s + 139-s + ⋯ |
| L(s) = 1 | − 0.894·5-s − 9-s + 1.20·11-s − 0.917·19-s + 1/5·25-s + 2.22·29-s + 5.02·31-s − 3.12·41-s + 0.894·45-s − 3/7·49-s − 1.07·55-s − 2.08·59-s + 0.512·61-s + 4.27·71-s + 0.900·79-s + 2/3·81-s + 1.27·89-s + 0.820·95-s − 1.20·99-s − 0.398·101-s − 4.21·109-s + 1.63·121-s − 0.715·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{6} \cdot 5^{6} \cdot 7^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{6} \cdot 5^{6} \cdot 7^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.312883337\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.312883337\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( ( 1 + T^{2} )^{3} \) |
| 5 | \( 1 + 2 T + 3 T^{2} + 12 T^{3} + 3 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 7 | \( ( 1 + T^{2} )^{3} \) |
| good | 11 | \( ( 1 - 2 T - 3 T^{2} + 60 T^{3} - 3 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 13 | \( 1 - 18 T^{2} - 25 T^{4} + 2404 T^{6} - 25 p^{2} T^{8} - 18 p^{4} T^{10} + p^{6} T^{12} \) |
| 17 | \( 1 + 26 T^{2} + 1007 T^{4} + 876 p T^{6} + 1007 p^{2} T^{8} + 26 p^{4} T^{10} + p^{6} T^{12} \) |
| 19 | \( ( 1 + 2 T - 3 T^{2} - 124 T^{3} - 3 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 23 | \( 1 + 6 T^{2} + 1151 T^{4} + 8788 T^{6} + 1151 p^{2} T^{8} + 6 p^{4} T^{10} + p^{6} T^{12} \) |
| 29 | \( ( 1 - 2 T + p T^{2} )^{6} \) |
| 31 | \( ( 1 - 14 T + 145 T^{2} - 908 T^{3} + 145 p T^{4} - 14 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 37 | \( 1 - 46 T^{2} + 2423 T^{4} - 52260 T^{6} + 2423 p^{2} T^{8} - 46 p^{4} T^{10} + p^{6} T^{12} \) |
| 41 | \( ( 1 + 10 T + 143 T^{2} + 812 T^{3} + 143 p T^{4} + 10 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 43 | \( 1 - 146 T^{2} + 11287 T^{4} - 585692 T^{6} + 11287 p^{2} T^{8} - 146 p^{4} T^{10} + p^{6} T^{12} \) |
| 47 | \( ( 1 - p T^{2} )^{6} \) |
| 53 | \( 1 - 226 T^{2} + 24215 T^{4} - 1593276 T^{6} + 24215 p^{2} T^{8} - 226 p^{4} T^{10} + p^{6} T^{12} \) |
| 59 | \( ( 1 + 8 T + 113 T^{2} + 688 T^{3} + 113 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 61 | \( ( 1 - 2 T - 29 T^{2} - 140 T^{3} - 29 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 67 | \( 1 - 146 T^{2} + 6919 T^{4} - 152348 T^{6} + 6919 p^{2} T^{8} - 146 p^{4} T^{10} + p^{6} T^{12} \) |
| 71 | \( ( 1 - 18 T + 161 T^{2} - 1204 T^{3} + 161 p T^{4} - 18 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 73 | \( 1 - 154 T^{2} + 2431 T^{4} + 519188 T^{6} + 2431 p^{2} T^{8} - 154 p^{4} T^{10} + p^{6} T^{12} \) |
| 79 | \( ( 1 - 4 T + 189 T^{2} - 568 T^{3} + 189 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 83 | \( 1 - 306 T^{2} + 47783 T^{4} - 4793948 T^{6} + 47783 p^{2} T^{8} - 306 p^{4} T^{10} + p^{6} T^{12} \) |
| 89 | \( ( 1 - 6 T + 143 T^{2} - 1300 T^{3} + 143 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 97 | \( 1 - 458 T^{2} + 97423 T^{4} - 12066764 T^{6} + 97423 p^{2} T^{8} - 458 p^{4} T^{10} + p^{6} T^{12} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{12} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−4.94900608211865650205103718124, −4.61897462494894681333250983222, −4.53228677761958922868746905640, −4.43686246709029978677315629674, −4.33017784552889387201559369052, −4.13077921822599670970705716554, −4.06470651024197752270995687351, −3.82582147463208900487057109343, −3.56202432856041743610982345722, −3.38129691002454990953104965376, −3.30987063563116237682606979886, −3.18745272688949554926810339142, −2.86349811258786289621682911177, −2.84820395713762471101419538693, −2.72872323708916814607659445729, −2.37971119372962361395838988519, −2.32418339933843958214773565752, −2.00503786719265638891576340206, −1.72951325100819977894398342628, −1.68771914612339091273721285716, −1.27109624379753640853342390219, −0.955003904261396408881979855788, −0.847910724087556684829637899944, −0.67470077047961518219938690437, −0.21429298850769099316222447182,
0.21429298850769099316222447182, 0.67470077047961518219938690437, 0.847910724087556684829637899944, 0.955003904261396408881979855788, 1.27109624379753640853342390219, 1.68771914612339091273721285716, 1.72951325100819977894398342628, 2.00503786719265638891576340206, 2.32418339933843958214773565752, 2.37971119372962361395838988519, 2.72872323708916814607659445729, 2.84820395713762471101419538693, 2.86349811258786289621682911177, 3.18745272688949554926810339142, 3.30987063563116237682606979886, 3.38129691002454990953104965376, 3.56202432856041743610982345722, 3.82582147463208900487057109343, 4.06470651024197752270995687351, 4.13077921822599670970705716554, 4.33017784552889387201559369052, 4.43686246709029978677315629674, 4.53228677761958922868746905640, 4.61897462494894681333250983222, 4.94900608211865650205103718124
Plot not available for L-functions of degree greater than 10.
