| L(s) = 1 | + 6·3-s − 2·4-s + 12·5-s − 10·7-s + 21·9-s − 12·11-s − 12·12-s + 72·15-s − 48·17-s − 42·19-s − 24·20-s − 60·21-s + 24·23-s + 58·25-s + 54·27-s + 20·28-s + 102·31-s − 72·33-s − 120·35-s − 42·36-s + 22·37-s + 28·43-s + 24·44-s + 252·45-s − 132·47-s + 49·49-s − 288·51-s + ⋯ |
| L(s) = 1 | + 2·3-s − 1/2·4-s + 12/5·5-s − 1.42·7-s + 7/3·9-s − 1.09·11-s − 12-s + 24/5·15-s − 2.82·17-s − 2.21·19-s − 6/5·20-s − 2.85·21-s + 1.04·23-s + 2.31·25-s + 2·27-s + 5/7·28-s + 3.29·31-s − 2.18·33-s − 3.42·35-s − 7/6·36-s + 0.594·37-s + 0.651·43-s + 6/11·44-s + 28/5·45-s − 2.80·47-s + 49-s − 5.64·51-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3111696 ^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3111696 ^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(2.286872514\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.286872514\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | $C_2^2$ | \( 1 + p T^{2} + p^{2} T^{4} \) |
| 3 | $C_2$ | \( ( 1 - p T + p T^{2} )^{2} \) |
| 7 | $C_2^2$ | \( 1 + 10 T + 51 T^{2} + 10 p^{2} T^{3} + p^{4} T^{4} \) |
| good | 5 | $D_4\times C_2$ | \( 1 - 12 T + 86 T^{2} - 456 T^{3} + 2019 T^{4} - 456 p^{2} T^{5} + 86 p^{4} T^{6} - 12 p^{6} T^{7} + p^{8} T^{8} \) |
| 11 | $C_2^2$ | \( ( 1 + 6 T - 85 T^{2} + 6 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 13 | $D_4\times C_2$ | \( 1 + 98 T^{2} + 54915 T^{4} + 98 p^{4} T^{6} + p^{8} T^{8} \) |
| 17 | $D_4\times C_2$ | \( 1 + 48 T + 1514 T^{2} + 35808 T^{3} + 694947 T^{4} + 35808 p^{2} T^{5} + 1514 p^{4} T^{6} + 48 p^{6} T^{7} + p^{8} T^{8} \) |
| 19 | $D_4\times C_2$ | \( 1 + 42 T + 1433 T^{2} + 35490 T^{3} + 795972 T^{4} + 35490 p^{2} T^{5} + 1433 p^{4} T^{6} + 42 p^{6} T^{7} + p^{8} T^{8} \) |
| 23 | $D_4\times C_2$ | \( 1 - 24 T + 22 T^{2} + 12096 T^{3} - 277629 T^{4} + 12096 p^{2} T^{5} + 22 p^{4} T^{6} - 24 p^{6} T^{7} + p^{8} T^{8} \) |
| 29 | $C_2^2$ | \( ( 1 + 530 T^{2} + p^{4} T^{4} )^{2} \) |
| 31 | $D_4\times C_2$ | \( 1 - 102 T + 6041 T^{2} - 8466 p T^{3} + 9396 p^{2} T^{4} - 8466 p^{3} T^{5} + 6041 p^{4} T^{6} - 102 p^{6} T^{7} + p^{8} T^{8} \) |
| 37 | $D_4\times C_2$ | \( 1 - 22 T - 2087 T^{2} + 3674 T^{3} + 4073284 T^{4} + 3674 p^{2} T^{5} - 2087 p^{4} T^{6} - 22 p^{6} T^{7} + p^{8} T^{8} \) |
| 41 | $D_4\times C_2$ | \( 1 - 2476 T^{2} + 6405414 T^{4} - 2476 p^{4} T^{6} + p^{8} T^{8} \) |
| 43 | $D_{4}$ | \( ( 1 - 14 T + 3675 T^{2} - 14 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 47 | $D_4\times C_2$ | \( 1 + 132 T + 11654 T^{2} + 771672 T^{3} + 42125907 T^{4} + 771672 p^{2} T^{5} + 11654 p^{4} T^{6} + 132 p^{6} T^{7} + p^{8} T^{8} \) |
| 53 | $D_4\times C_2$ | \( 1 - 120 T + 110 p T^{2} - 354240 T^{3} + 25104819 T^{4} - 354240 p^{2} T^{5} + 110 p^{5} T^{6} - 120 p^{6} T^{7} + p^{8} T^{8} \) |
| 59 | $D_4\times C_2$ | \( 1 + 24 T + 6026 T^{2} + 140016 T^{3} + 22586547 T^{4} + 140016 p^{2} T^{5} + 6026 p^{4} T^{6} + 24 p^{6} T^{7} + p^{8} T^{8} \) |
| 61 | $D_4\times C_2$ | \( 1 + 72 T + 9218 T^{2} + 539280 T^{3} + 48684147 T^{4} + 539280 p^{2} T^{5} + 9218 p^{4} T^{6} + 72 p^{6} T^{7} + p^{8} T^{8} \) |
| 67 | $D_4\times C_2$ | \( 1 - 110 T + 3625 T^{2} + 55330 T^{3} - 2642396 T^{4} + 55330 p^{2} T^{5} + 3625 p^{4} T^{6} - 110 p^{6} T^{7} + p^{8} T^{8} \) |
| 71 | $D_{4}$ | \( ( 1 - 156 T + 178 p T^{2} - 156 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 73 | $D_4\times C_2$ | \( 1 + 66 T + 2873 T^{2} + 93786 T^{3} - 18641292 T^{4} + 93786 p^{2} T^{5} + 2873 p^{4} T^{6} + 66 p^{6} T^{7} + p^{8} T^{8} \) |
| 79 | $D_4\times C_2$ | \( 1 + 10 T - 3695 T^{2} - 86870 T^{3} - 25172156 T^{4} - 86870 p^{2} T^{5} - 3695 p^{4} T^{6} + 10 p^{6} T^{7} + p^{8} T^{8} \) |
| 83 | $D_4\times C_2$ | \( 1 - 116 p T^{2} + 107694438 T^{4} - 116 p^{5} T^{6} + p^{8} T^{8} \) |
| 89 | $C_2^2$ | \( ( 1 + 36 T + 8353 T^{2} + 36 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 97 | $D_4\times C_2$ | \( 1 - 36580 T^{2} + 511416774 T^{4} - 36580 p^{4} T^{6} + p^{8} T^{8} \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−12.10140574466711341635237200530, −11.12821572458445703318137204726, −11.09659642757673927786941720914, −10.59955010809873457371118535736, −10.41022451160148627894738851518, −9.815225387421475556026287969762, −9.743878526286321490894530839445, −9.530279792776025170513439817410, −9.346296849718374923314698171812, −8.768268876148122137929799949480, −8.453261571995771634331487690457, −8.364627548432850734657708634602, −8.118739712697830033142099782363, −7.20074082677624703085269624357, −6.71536788738059700592190572363, −6.56597622883056463986598583795, −6.35902541952197033626076331665, −5.85989696359579339620966549555, −5.15068531751407392523279398199, −4.51522153840729080074153825979, −4.42902515379082659072706649281, −3.52432439487018561702569581882, −2.72031556201279849766172301524, −2.36411800528140250691883631837, −2.20604359483068962765831611063,
2.20604359483068962765831611063, 2.36411800528140250691883631837, 2.72031556201279849766172301524, 3.52432439487018561702569581882, 4.42902515379082659072706649281, 4.51522153840729080074153825979, 5.15068531751407392523279398199, 5.85989696359579339620966549555, 6.35902541952197033626076331665, 6.56597622883056463986598583795, 6.71536788738059700592190572363, 7.20074082677624703085269624357, 8.118739712697830033142099782363, 8.364627548432850734657708634602, 8.453261571995771634331487690457, 8.768268876148122137929799949480, 9.346296849718374923314698171812, 9.530279792776025170513439817410, 9.743878526286321490894530839445, 9.815225387421475556026287969762, 10.41022451160148627894738851518, 10.59955010809873457371118535736, 11.09659642757673927786941720914, 11.12821572458445703318137204726, 12.10140574466711341635237200530
