| L(s) = 1 | − 6·2-s + 21·4-s + 2·5-s − 10·7-s − 56·8-s + 10·9-s − 12·10-s + 60·14-s + 126·16-s − 60·18-s + 42·20-s + 25-s − 210·28-s + 28·29-s − 252·32-s − 20·35-s + 210·36-s − 10·37-s − 112·40-s + 20·45-s − 14·47-s + 27·49-s − 6·50-s + 560·56-s − 168·58-s + 8·61-s − 100·63-s + ⋯ |
| L(s) = 1 | − 4.24·2-s + 21/2·4-s + 0.894·5-s − 3.77·7-s − 19.7·8-s + 10/3·9-s − 3.79·10-s + 16.0·14-s + 63/2·16-s − 14.1·18-s + 9.39·20-s + 1/5·25-s − 39.6·28-s + 5.19·29-s − 44.5·32-s − 3.38·35-s + 35·36-s − 1.64·37-s − 17.7·40-s + 2.98·45-s − 2.04·47-s + 27/7·49-s − 0.848·50-s + 74.8·56-s − 22.0·58-s + 1.02·61-s − 12.5·63-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{6} \cdot 5^{6} \cdot 13^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{6} \cdot 5^{6} \cdot 13^{12}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.08659414707\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.08659414707\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( ( 1 + T )^{6} \) |
| 5 | \( 1 - 2 T + 3 T^{2} - 12 T^{3} + 3 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 13 | \( 1 \) |
| good | 3 | \( 1 - 10 T^{2} + 55 T^{4} - 200 T^{6} + 55 p^{2} T^{8} - 10 p^{4} T^{10} + p^{6} T^{12} \) |
| 7 | \( ( 1 + 5 T + 24 T^{2} + 65 T^{3} + 24 p T^{4} + 5 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 11 | \( 1 - 3 T^{2} - 42 T^{4} - 51 T^{6} - 42 p^{2} T^{8} - 3 p^{4} T^{10} + p^{6} T^{12} \) |
| 17 | \( 1 - 90 T^{2} + 3539 T^{4} - 78128 T^{6} + 3539 p^{2} T^{8} - 90 p^{4} T^{10} + p^{6} T^{12} \) |
| 19 | \( 1 - 31 T^{2} + 826 T^{4} - 22699 T^{6} + 826 p^{2} T^{8} - 31 p^{4} T^{10} + p^{6} T^{12} \) |
| 23 | \( 1 - 58 T^{2} + 2175 T^{4} - 57244 T^{6} + 2175 p^{2} T^{8} - 58 p^{4} T^{10} + p^{6} T^{12} \) |
| 29 | \( ( 1 - 14 T + 115 T^{2} - 660 T^{3} + 115 p T^{4} - 14 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 31 | \( 1 - 34 T^{2} + 1547 T^{4} - 42624 T^{6} + 1547 p^{2} T^{8} - 34 p^{4} T^{10} + p^{6} T^{12} \) |
| 37 | \( ( 1 + 5 T + 82 T^{2} + 263 T^{3} + 82 p T^{4} + 5 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 41 | \( 1 - 154 T^{2} + 12143 T^{4} - 615420 T^{6} + 12143 p^{2} T^{8} - 154 p^{4} T^{10} + p^{6} T^{12} \) |
| 43 | \( ( 1 - 82 T^{2} + p^{2} T^{4} )^{3} \) |
| 47 | \( ( 1 + 7 T + 152 T^{2} + 659 T^{3} + 152 p T^{4} + 7 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 53 | \( 1 - 223 T^{2} + 22278 T^{4} - 1406239 T^{6} + 22278 p^{2} T^{8} - 223 p^{4} T^{10} + p^{6} T^{12} \) |
| 59 | \( 1 - 226 T^{2} + 27383 T^{4} - 1994556 T^{6} + 27383 p^{2} T^{8} - 226 p^{4} T^{10} + p^{6} T^{12} \) |
| 61 | \( ( 1 - 4 T + 75 T^{2} + 2 p T^{3} + 75 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 67 | \( ( 1 - 12 T + 233 T^{2} - 1624 T^{3} + 233 p T^{4} - 12 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 71 | \( 1 - 106 T^{2} + 4895 T^{4} - 126924 T^{6} + 4895 p^{2} T^{8} - 106 p^{4} T^{10} + p^{6} T^{12} \) |
| 73 | \( ( 1 - 6 T + 93 T^{2} - 1146 T^{3} + 93 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 79 | \( ( 1 - 14 T + 195 T^{2} - 26 p T^{3} + 195 p T^{4} - 14 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 83 | \( ( 1 + 4 T + 121 T^{2} + 618 T^{3} + 121 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 89 | \( 1 - 499 T^{2} + 106670 T^{4} - 12491679 T^{6} + 106670 p^{2} T^{8} - 499 p^{4} T^{10} + p^{6} T^{12} \) |
| 97 | \( ( 1 - 26 T + 493 T^{2} - 5510 T^{3} + 493 p T^{4} - 26 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 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
−4.83651936679273833082209180201, −4.83266434705224828979662591277, −4.66461302625224581899076836275, −4.53106553678898542915903974020, −4.17172853174561242905277696618, −3.79673079014788672793043125333, −3.70874744244873734117679474424, −3.69122931110142317434425938838, −3.52269126003654965343336890930, −3.37063464861288702545848381952, −3.12243451133379677377683200291, −2.91503763116634501833026959760, −2.90454607457234182946814426110, −2.56708475259988922705508213868, −2.55701275998523786572722661866, −2.08559202767458613479225113556, −2.06864663704107336271342144870, −1.93126579864349414767906196286, −1.71856192334079197667671739584, −1.33512436369982654431629062071, −1.23395460555189761127149474105, −0.894808874765089164190456505707, −0.78104320247612924698086793242, −0.68737132230390713739378314963, −0.096587643302854067559052822190,
0.096587643302854067559052822190, 0.68737132230390713739378314963, 0.78104320247612924698086793242, 0.894808874765089164190456505707, 1.23395460555189761127149474105, 1.33512436369982654431629062071, 1.71856192334079197667671739584, 1.93126579864349414767906196286, 2.06864663704107336271342144870, 2.08559202767458613479225113556, 2.55701275998523786572722661866, 2.56708475259988922705508213868, 2.90454607457234182946814426110, 2.91503763116634501833026959760, 3.12243451133379677377683200291, 3.37063464861288702545848381952, 3.52269126003654965343336890930, 3.69122931110142317434425938838, 3.70874744244873734117679474424, 3.79673079014788672793043125333, 4.17172853174561242905277696618, 4.53106553678898542915903974020, 4.66461302625224581899076836275, 4.83266434705224828979662591277, 4.83651936679273833082209180201
Plot not available for L-functions of degree greater than 10.
