L(s) = 1 | − 2·5-s − 8·7-s + 16-s − 2·17-s − 2·19-s + 2·23-s + 25-s + 16·35-s + 2·43-s − 2·47-s + 37·49-s + 2·73-s − 2·80-s + 81-s − 2·83-s + 4·85-s + 4·95-s − 8·112-s − 4·115-s + 16·119-s + 121-s + 127-s + 131-s + 16·133-s + 137-s + 139-s + 149-s + ⋯ |
L(s) = 1 | − 2·5-s − 8·7-s + 16-s − 2·17-s − 2·19-s + 2·23-s + 25-s + 16·35-s + 2·43-s − 2·47-s + 37·49-s + 2·73-s − 2·80-s + 81-s − 2·83-s + 4·85-s + 4·95-s − 8·112-s − 4·115-s + 16·119-s + 121-s + 127-s + 131-s + 16·133-s + 137-s + 139-s + 149-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{8} \cdot 11^{8} \cdot 19^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{8} \cdot 11^{8} \cdot 19^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.004065253934\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.004065253934\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \) |
| 11 | \( 1 - T^{2} + T^{4} - T^{6} + T^{8} \) |
| 19 | \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \) |
good | 2 | \( 1 - T^{4} + T^{8} - T^{12} + T^{16} \) |
| 3 | \( 1 - T^{4} + T^{8} - T^{12} + T^{16} \) |
| 7 | \( ( 1 + T )^{8}( 1 - T^{2} + T^{4} - T^{6} + T^{8} ) \) |
| 13 | \( 1 - T^{4} + T^{8} - T^{12} + T^{16} \) |
| 17 | \( ( 1 + T^{2} )^{4}( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \) |
| 23 | \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2}( 1 - T^{2} + T^{4} - T^{6} + T^{8} ) \) |
| 29 | \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2}( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \) |
| 31 | \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2}( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \) |
| 37 | \( 1 - T^{4} + T^{8} - T^{12} + T^{16} \) |
| 41 | \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{2} \) |
| 43 | \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2}( 1 - T^{2} + T^{4} - T^{6} + T^{8} ) \) |
| 47 | \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{2}( 1 - T^{2} + T^{4} - T^{6} + T^{8} ) \) |
| 53 | \( 1 - T^{4} + T^{8} - T^{12} + T^{16} \) |
| 59 | \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{2} \) |
| 61 | \( ( 1 + T^{2} )^{4}( 1 - T^{2} + T^{4} - T^{6} + T^{8} ) \) |
| 67 | \( ( 1 + T^{4} )^{4} \) |
| 71 | \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2}( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \) |
| 73 | \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2}( 1 - T^{2} + T^{4} - T^{6} + T^{8} ) \) |
| 79 | \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2}( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \) |
| 83 | \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{2}( 1 - T^{2} + T^{4} - T^{6} + T^{8} ) \) |
| 89 | \( ( 1 + T^{2} )^{8} \) |
| 97 | \( 1 - T^{4} + T^{8} - T^{12} + T^{16} \) |
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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
−4.30086756444118193526864531424, −4.18044667961218228684078735921, −4.08983570414524206480266263171, −4.02053273824161051808262000806, −3.91821926641694518179665960016, −3.88961080248027901762942480075, −3.58272328720687667997332056707, −3.52426926814187165538719840398, −3.37392278937676019395858358822, −3.34744496540404984192603111420, −3.20525080269204547743481928837, −3.17257571266069514786330980794, −3.08204040309303206533053353630, −2.75210331790125772987230562119, −2.74682795875298409837362136743, −2.36589506210677997749271771848, −2.31498609638475493110199832038, −2.27400980854831912583511349420, −2.21840498176827732095871074766, −2.20167155601630569231964382076, −1.26592378844397844517887311359, −1.09912894912133402956015690423, −1.08352969010111989399973448756, −0.59207095733602129139067829226, −0.083150517460411159550230369345,
0.083150517460411159550230369345, 0.59207095733602129139067829226, 1.08352969010111989399973448756, 1.09912894912133402956015690423, 1.26592378844397844517887311359, 2.20167155601630569231964382076, 2.21840498176827732095871074766, 2.27400980854831912583511349420, 2.31498609638475493110199832038, 2.36589506210677997749271771848, 2.74682795875298409837362136743, 2.75210331790125772987230562119, 3.08204040309303206533053353630, 3.17257571266069514786330980794, 3.20525080269204547743481928837, 3.34744496540404984192603111420, 3.37392278937676019395858358822, 3.52426926814187165538719840398, 3.58272328720687667997332056707, 3.88961080248027901762942480075, 3.91821926641694518179665960016, 4.02053273824161051808262000806, 4.08983570414524206480266263171, 4.18044667961218228684078735921, 4.30086756444118193526864531424
Plot not available for L-functions of degree greater than 10.