| L(s) = 1 | − 2·3-s + 3·4-s + 2·9-s − 6·12-s + 2·13-s + 3·16-s − 4·23-s + 4·25-s − 8·29-s + 6·36-s − 4·39-s + 2·41-s + 2·47-s − 6·48-s + 6·52-s − 8·59-s − 2·64-s + 8·69-s − 2·71-s − 8·75-s − 81-s + 16·87-s − 12·92-s + 12·100-s − 4·101-s − 24·116-s + 4·117-s + ⋯ |
| L(s) = 1 | − 2·3-s + 3·4-s + 2·9-s − 6·12-s + 2·13-s + 3·16-s − 4·23-s + 4·25-s − 8·29-s + 6·36-s − 4·39-s + 2·41-s + 2·47-s − 6·48-s + 6·52-s − 8·59-s − 2·64-s + 8·69-s − 2·71-s − 8·75-s − 81-s + 16·87-s − 12·92-s + 12·100-s − 4·101-s − 24·116-s + 4·117-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(23^{16} \cdot 41^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(23^{16} \cdot 41^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.2407293837\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2407293837\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 23 | \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{4} \) |
| 41 | \( ( 1 - T + T^{3} - T^{4} + T^{5} - T^{7} + T^{8} )^{2} \) |
| good | 2 | \( ( 1 - T^{2} + T^{4} )^{4}( 1 + T^{2} - T^{6} - T^{8} - T^{10} + T^{14} + T^{16} ) \) |
| 3 | \( ( 1 + T - T^{3} - T^{4} - T^{5} + T^{7} + T^{8} )^{2}( 1 + T^{2} - T^{6} - T^{8} - T^{10} + T^{14} + T^{16} ) \) |
| 5 | \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{4} \) |
| 7 | \( ( 1 - T^{4} + T^{8} - T^{12} + T^{16} )^{2} \) |
| 11 | \( ( 1 - T^{4} + T^{8} - T^{12} + T^{16} )^{2} \) |
| 13 | \( ( 1 - T + T^{3} - T^{4} + T^{5} - T^{7} + T^{8} )^{2}( 1 + T^{2} - T^{6} - T^{8} - T^{10} + T^{14} + T^{16} ) \) |
| 17 | \( ( 1 - T^{4} + T^{8} - T^{12} + T^{16} )^{2} \) |
| 19 | \( ( 1 - T^{4} + T^{8} - T^{12} + T^{16} )^{2} \) |
| 29 | \( ( 1 + T + T^{2} )^{8}( 1 + T^{2} - T^{6} - T^{8} - T^{10} + T^{14} + T^{16} ) \) |
| 31 | \( ( 1 - T^{2} + T^{4} )^{4}( 1 + T^{2} - T^{6} - T^{8} - T^{10} + T^{14} + T^{16} ) \) |
| 37 | \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{4}( 1 + T + T^{2} + T^{3} + T^{4} )^{4} \) |
| 43 | \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{4} \) |
| 47 | \( ( 1 - T^{2} + T^{4} )^{4}( 1 - T + T^{3} - T^{4} + T^{5} - T^{7} + T^{8} )^{2} \) |
| 53 | \( ( 1 - T^{4} + T^{8} - T^{12} + T^{16} )^{2} \) |
| 59 | \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{8} \) |
| 61 | \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{4} \) |
| 67 | \( ( 1 - T^{4} + T^{8} - T^{12} + T^{16} )^{2} \) |
| 71 | \( ( 1 + T - T^{3} - T^{4} - T^{5} + T^{7} + T^{8} )^{2}( 1 + T^{2} - T^{6} - T^{8} - T^{10} + T^{14} + T^{16} ) \) |
| 73 | \( ( 1 - T + T^{3} - T^{4} + T^{5} - T^{7} + T^{8} )^{2}( 1 + T - T^{3} - T^{4} - T^{5} + T^{7} + T^{8} )^{2} \) |
| 79 | \( ( 1 + T^{4} )^{8} \) |
| 83 | \( ( 1 - T )^{16}( 1 + T )^{16} \) |
| 89 | \( ( 1 - T^{4} + T^{8} - T^{12} + T^{16} )^{2} \) |
| 97 | \( ( 1 - T^{4} + T^{8} - T^{12} + T^{16} )^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{32} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−2.89000792570922135211314014994, −2.82008836860571423314636731758, −2.79332280288089708555501706200, −2.69811017509965093664790945403, −2.59844099779268342328565780217, −2.54830628156530267746049712925, −2.51359092718453813592786809355, −2.27492893653081643326028451790, −2.26550828896855368535095418521, −2.12042993710964606406707275351, −2.00085295430743075985772892434, −1.88887943844739407567196683642, −1.86765192223819035797346627825, −1.84231592711039182908169170396, −1.76753462718111631334388473698, −1.65512285823835893651206760768, −1.62088859145956747480788306046, −1.53451145443997841494593188144, −1.50679053844028745711695509570, −1.33407097231475687058636465056, −1.16689033925713311898573111436, −1.11246441615025548947746899421, −1.00765547252638300073012316760, −0.75855776052480419307337607101, −0.38354315758073013140926621289,
0.38354315758073013140926621289, 0.75855776052480419307337607101, 1.00765547252638300073012316760, 1.11246441615025548947746899421, 1.16689033925713311898573111436, 1.33407097231475687058636465056, 1.50679053844028745711695509570, 1.53451145443997841494593188144, 1.62088859145956747480788306046, 1.65512285823835893651206760768, 1.76753462718111631334388473698, 1.84231592711039182908169170396, 1.86765192223819035797346627825, 1.88887943844739407567196683642, 2.00085295430743075985772892434, 2.12042993710964606406707275351, 2.26550828896855368535095418521, 2.27492893653081643326028451790, 2.51359092718453813592786809355, 2.54830628156530267746049712925, 2.59844099779268342328565780217, 2.69811017509965093664790945403, 2.79332280288089708555501706200, 2.82008836860571423314636731758, 2.89000792570922135211314014994
Plot not available for L-functions of degree greater than 10.
