L(s) = 1 | − 2·7-s + 4·9-s + 4·11-s − 4·23-s − 25-s − 6·29-s − 2·37-s − 2·43-s + 49-s − 8·63-s − 8·77-s + 8·79-s + 6·81-s + 16·99-s + 6·107-s + 4·109-s + 6·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 8·161-s + 163-s + 167-s + ⋯ |
L(s) = 1 | − 2·7-s + 4·9-s + 4·11-s − 4·23-s − 25-s − 6·29-s − 2·37-s − 2·43-s + 49-s − 8·63-s − 8·77-s + 8·79-s + 6·81-s + 16·99-s + 6·107-s + 4·109-s + 6·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 8·161-s + 163-s + 167-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 7^{8} \cdot 41^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 7^{8} \cdot 41^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.9281073344\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9281073344\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 7 | \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \) |
| 41 | \( 1 - T^{2} + T^{4} - T^{6} + T^{8} \) |
good | 3 | \( ( 1 - T^{2} + T^{4} )^{4} \) |
| 5 | \( 1 + T^{2} - T^{6} - T^{8} - T^{10} + T^{14} + T^{16} \) |
| 11 | \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{4} \) |
| 13 | \( 1 + T^{2} - T^{6} - T^{8} - T^{10} + T^{14} + T^{16} \) |
| 17 | \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{2} \) |
| 19 | \( 1 + T^{2} - T^{6} - T^{8} - T^{10} + T^{14} + T^{16} \) |
| 23 | \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{4} \) |
| 29 | \( ( 1 + T )^{8}( 1 - T + T^{2} - T^{3} + T^{4} )^{2} \) |
| 31 | \( ( 1 + T^{2} )^{4}( 1 - T^{2} + T^{4} - T^{6} + T^{8} ) \) |
| 37 | \( ( 1 + T - T^{3} - T^{4} - T^{5} + T^{7} + T^{8} )^{2} \) |
| 43 | \( ( 1 + T - T^{3} - T^{4} - T^{5} + T^{7} + T^{8} )^{2} \) |
| 47 | \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{2} \) |
| 53 | \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{2} \) |
| 59 | \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{2} \) |
| 61 | \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{2} \) |
| 67 | \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{2} \) |
| 71 | \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{2} \) |
| 73 | \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{2} \) |
| 79 | \( ( 1 - T + T^{2} )^{8} \) |
| 83 | \( ( 1 - T^{2} + T^{4} )^{4} \) |
| 89 | \( ( 1 + T^{2} )^{4}( 1 - T^{2} + T^{4} - T^{6} + T^{8} ) \) |
| 97 | \( 1 + T^{2} - T^{6} - T^{8} - T^{10} + T^{14} + T^{16} \) |
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
−4.33502516858585723191863004225, −4.19860332853286735152006471386, −4.10348167578830668093049217838, −3.92701632411256822637741222932, −3.84204859307483786844571902384, −3.76989916261071248229868798099, −3.64648862564269952157790599424, −3.63110088906770882665329424562, −3.46689244697828483856567413030, −3.44922150668446790820367215838, −3.33727996604478259022845221442, −3.21071544385479668746564437921, −3.14445561347690398015634708095, −2.42617748776200514945094350410, −2.37589993944782810652193623903, −2.12432984687920650746321530622, −2.04776698668677107267462544796, −1.98748383566477736721312006568, −1.90442345253304963982786053466, −1.73059664554291718060909411395, −1.68063720069178119651450381867, −1.45851542342228874076480281989, −1.29128018708751784883325961845, −0.913343631389713829830884171142, −0.59816032920170460789373094072,
0.59816032920170460789373094072, 0.913343631389713829830884171142, 1.29128018708751784883325961845, 1.45851542342228874076480281989, 1.68063720069178119651450381867, 1.73059664554291718060909411395, 1.90442345253304963982786053466, 1.98748383566477736721312006568, 2.04776698668677107267462544796, 2.12432984687920650746321530622, 2.37589993944782810652193623903, 2.42617748776200514945094350410, 3.14445561347690398015634708095, 3.21071544385479668746564437921, 3.33727996604478259022845221442, 3.44922150668446790820367215838, 3.46689244697828483856567413030, 3.63110088906770882665329424562, 3.64648862564269952157790599424, 3.76989916261071248229868798099, 3.84204859307483786844571902384, 3.92701632411256822637741222932, 4.10348167578830668093049217838, 4.19860332853286735152006471386, 4.33502516858585723191863004225
Plot not available for L-functions of degree greater than 10.