| L(s) = 1 | − 3·4-s + 3·9-s + 6·16-s + 6·29-s − 18·31-s − 9·36-s − 24·41-s + 3·49-s + 30·59-s + 30·61-s − 10·64-s − 72·71-s − 18·79-s − 3·81-s − 6·101-s − 72·109-s − 18·116-s − 18·121-s + 54·124-s + 127-s + 131-s + 137-s + 139-s + 18·144-s + 149-s + 151-s + 157-s + ⋯ |
| L(s) = 1 | − 3/2·4-s + 9-s + 3/2·16-s + 1.11·29-s − 3.23·31-s − 3/2·36-s − 3.74·41-s + 3/7·49-s + 3.90·59-s + 3.84·61-s − 5/4·64-s − 8.54·71-s − 2.02·79-s − 1/3·81-s − 0.597·101-s − 6.89·109-s − 1.67·116-s − 1.63·121-s + 4.84·124-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 3/2·144-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{6} \cdot 5^{12} \cdot 29^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{6} \cdot 5^{12} \cdot 29^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9505199400\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9505199400\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( ( 1 + T^{2} )^{3} \) |
| 5 | \( 1 \) |
| 29 | \( ( 1 - T )^{6} \) |
| good | 3 | \( 1 - p T^{2} + 4 p T^{4} - 8 T^{6} + 4 p^{3} T^{8} - p^{5} T^{10} + p^{6} T^{12} \) |
| 7 | \( 1 - 3 T^{2} + 144 T^{4} - 292 T^{6} + 144 p^{2} T^{8} - 3 p^{4} T^{10} + p^{6} T^{12} \) |
| 11 | \( ( 1 + 9 T^{2} - 24 T^{3} + 9 p T^{4} + p^{3} T^{6} )^{2} \) |
| 13 | \( 1 - 3 T^{2} + 432 T^{4} - 688 T^{6} + 432 p^{2} T^{8} - 3 p^{4} T^{10} + p^{6} T^{12} \) |
| 17 | \( 1 - 75 T^{2} + 2688 T^{4} - 57544 T^{6} + 2688 p^{2} T^{8} - 75 p^{4} T^{10} + p^{6} T^{12} \) |
| 19 | \( ( 1 + 9 T^{2} + 56 T^{3} + 9 p T^{4} + p^{3} T^{6} )^{2} \) |
| 23 | \( 1 + 9 T^{2} + 72 T^{4} - 788 p T^{6} + 72 p^{2} T^{8} + 9 p^{4} T^{10} + p^{6} T^{12} \) |
| 31 | \( ( 1 + 9 T + 108 T^{2} + 556 T^{3} + 108 p T^{4} + 9 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 37 | \( 1 - 78 T^{2} + 4407 T^{4} - 160036 T^{6} + 4407 p^{2} T^{8} - 78 p^{4} T^{10} + p^{6} T^{12} \) |
| 41 | \( ( 1 + 12 T + 99 T^{2} + 528 T^{3} + 99 p T^{4} + 12 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 43 | \( 1 - 207 T^{2} + 19476 T^{4} - 1068208 T^{6} + 19476 p^{2} T^{8} - 207 p^{4} T^{10} + p^{6} T^{12} \) |
| 47 | \( 1 - 90 T^{2} + 6255 T^{4} - 361132 T^{6} + 6255 p^{2} T^{8} - 90 p^{4} T^{10} + p^{6} T^{12} \) |
| 53 | \( 1 - 75 T^{2} + 5928 T^{4} - 268576 T^{6} + 5928 p^{2} T^{8} - 75 p^{4} T^{10} + p^{6} T^{12} \) |
| 59 | \( ( 1 - 15 T + 180 T^{2} - 1602 T^{3} + 180 p T^{4} - 15 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 61 | \( ( 1 - 15 T + 204 T^{2} - 1604 T^{3} + 204 p T^{4} - 15 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 67 | \( 1 - 78 T^{2} + 6999 T^{4} - 297412 T^{6} + 6999 p^{2} T^{8} - 78 p^{4} T^{10} + p^{6} T^{12} \) |
| 71 | \( ( 1 + 12 T + p T^{2} )^{6} \) |
| 73 | \( 1 - 387 T^{2} + 65304 T^{4} - 6187768 T^{6} + 65304 p^{2} T^{8} - 387 p^{4} T^{10} + p^{6} T^{12} \) |
| 79 | \( ( 1 + 9 T + 102 T^{2} + 320 T^{3} + 102 p T^{4} + 9 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 83 | \( 1 - 354 T^{2} + 57255 T^{4} - 5765308 T^{6} + 57255 p^{2} T^{8} - 354 p^{4} T^{10} + p^{6} T^{12} \) |
| 89 | \( ( 1 + 243 T^{2} + 24 T^{3} + 243 p T^{4} + p^{3} T^{6} )^{2} \) |
| 97 | \( 1 - 483 T^{2} + 102768 T^{4} - 12673240 T^{6} + 102768 p^{2} T^{8} - 483 p^{4} T^{10} + p^{6} T^{12} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{12} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−4.99536767283335832775098899439, −4.90787813651172564238040722425, −4.71735237403136486583855985879, −4.52330650544540243583152532362, −4.15594265487687509550592989957, −4.14183583460925825183745621764, −4.09847792879907851103873961561, −3.86843238239122745072035765538, −3.83800649250653800171664323632, −3.82136970534892586766996473940, −3.36511465977222818579666519041, −3.35021983587329677240840744518, −2.89241877728842826427248721066, −2.86804476334973866367887780305, −2.73115465683998882377577894182, −2.54940931302745274511644409089, −2.40646029926149221049358069514, −1.82389367743632713457765240200, −1.72043373027174672904428374783, −1.53235544852342953128186942019, −1.50288029230666707008313457400, −1.29585052507668996924407392497, −0.886800886204523991175676590830, −0.31823205749069058925732344679, −0.26659252343118219712424611478,
0.26659252343118219712424611478, 0.31823205749069058925732344679, 0.886800886204523991175676590830, 1.29585052507668996924407392497, 1.50288029230666707008313457400, 1.53235544852342953128186942019, 1.72043373027174672904428374783, 1.82389367743632713457765240200, 2.40646029926149221049358069514, 2.54940931302745274511644409089, 2.73115465683998882377577894182, 2.86804476334973866367887780305, 2.89241877728842826427248721066, 3.35021983587329677240840744518, 3.36511465977222818579666519041, 3.82136970534892586766996473940, 3.83800649250653800171664323632, 3.86843238239122745072035765538, 4.09847792879907851103873961561, 4.14183583460925825183745621764, 4.15594265487687509550592989957, 4.52330650544540243583152532362, 4.71735237403136486583855985879, 4.90787813651172564238040722425, 4.99536767283335832775098899439
Plot not available for L-functions of degree greater than 10.
