L(s) = 1 | + 2·2-s + 4-s − 8·7-s − 3·9-s − 16·14-s − 6·18-s − 4·23-s + 25-s − 8·28-s − 2·32-s − 3·36-s − 8·46-s + 36·49-s + 2·50-s + 24·63-s − 4·64-s + 4·71-s − 6·79-s + 6·81-s − 4·92-s + 72·98-s + 100-s − 6·113-s − 2·121-s + 48·126-s + 127-s − 2·128-s + ⋯ |
L(s) = 1 | + 2·2-s + 4-s − 8·7-s − 3·9-s − 16·14-s − 6·18-s − 4·23-s + 25-s − 8·28-s − 2·32-s − 3·36-s − 8·46-s + 36·49-s + 2·50-s + 24·63-s − 4·64-s + 4·71-s − 6·79-s + 6·81-s − 4·92-s + 72·98-s + 100-s − 6·113-s − 2·121-s + 48·126-s + 127-s − 2·128-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 5^{16} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 5^{16} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.02886826731\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.02886826731\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2} \) |
| 5 | \( 1 - T^{2} + T^{4} - T^{6} + T^{8} \) |
| 7 | \( ( 1 + T )^{8} \) |
good | 3 | \( ( 1 + T^{2} )^{4}( 1 - T^{2} + T^{4} - T^{6} + T^{8} ) \) |
| 11 | \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2}( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \) |
| 13 | \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{2} \) |
| 17 | \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2}( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \) |
| 19 | \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{2} \) |
| 23 | \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{4} \) |
| 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 | \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2}( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \) |
| 41 | \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2}( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \) |
| 43 | \( ( 1 - T )^{8}( 1 + T )^{8} \) |
| 47 | \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2}( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \) |
| 53 | \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2}( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \) |
| 59 | \( ( 1 + T^{2} )^{4}( 1 - T^{2} + T^{4} - T^{6} + T^{8} ) \) |
| 61 | \( ( 1 + T^{2} )^{4}( 1 - T^{2} + T^{4} - T^{6} + T^{8} ) \) |
| 67 | \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2}( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \) |
| 71 | \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{4} \) |
| 73 | \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2}( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \) |
| 79 | \( ( 1 + T )^{8}( 1 - T + T^{2} - T^{3} + T^{4} )^{2} \) |
| 83 | \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{2} \) |
| 89 | \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2}( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \) |
| 97 | \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2}( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{16} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−4.19190532648003138063654156884, −4.04464906417913555312352078416, −4.00035878326675919081495598049, −3.99046165098812142903869231700, −3.82758548451803870063428620526, −3.59567777942205648497474576155, −3.49943729842409964607718231947, −3.48607617149681610483187808566, −3.25838588822926833516245191684, −3.25653215320504421441930454776, −3.12310501507340961386727973010, −2.87030431347553933941278850483, −2.84926056531156137955195571858, −2.80773861137065744452294652644, −2.58009072561909172004609424629, −2.43838579966832259913146502127, −2.35155292541466349688018106358, −2.22733079856664695251102372414, −2.22580581041697133799386198089, −1.86298532243176376511015882646, −1.38374054779614435746599674834, −1.12233116392352343792814609319, −1.04958346167430216007517007516, −0.26003480689122438151243226334, −0.21081003325642543076509382466,
0.21081003325642543076509382466, 0.26003480689122438151243226334, 1.04958346167430216007517007516, 1.12233116392352343792814609319, 1.38374054779614435746599674834, 1.86298532243176376511015882646, 2.22580581041697133799386198089, 2.22733079856664695251102372414, 2.35155292541466349688018106358, 2.43838579966832259913146502127, 2.58009072561909172004609424629, 2.80773861137065744452294652644, 2.84926056531156137955195571858, 2.87030431347553933941278850483, 3.12310501507340961386727973010, 3.25653215320504421441930454776, 3.25838588822926833516245191684, 3.48607617149681610483187808566, 3.49943729842409964607718231947, 3.59567777942205648497474576155, 3.82758548451803870063428620526, 3.99046165098812142903869231700, 4.00035878326675919081495598049, 4.04464906417913555312352078416, 4.19190532648003138063654156884
Plot not available for L-functions of degree greater than 10.