| L(s) = 1 | + 4·3-s + 2·9-s − 16·13-s + 25-s − 12·27-s − 16·31-s − 16·37-s − 64·39-s − 4·41-s − 28·43-s + 18·49-s + 32·53-s − 20·67-s − 24·71-s + 4·75-s + 32·79-s − 19·81-s + 60·83-s − 20·89-s − 64·93-s − 20·107-s − 64·111-s − 32·117-s + 22·121-s − 16·123-s + 16·125-s + 127-s + ⋯ |
| L(s) = 1 | + 2.30·3-s + 2/3·9-s − 4.43·13-s + 1/5·25-s − 2.30·27-s − 2.87·31-s − 2.63·37-s − 10.2·39-s − 0.624·41-s − 4.26·43-s + 18/7·49-s + 4.39·53-s − 2.44·67-s − 2.84·71-s + 0.461·75-s + 3.60·79-s − 2.11·81-s + 6.58·83-s − 2.11·89-s − 6.63·93-s − 1.93·107-s − 6.07·111-s − 2.95·117-s + 2·121-s − 1.44·123-s + 1.43·125-s + 0.0887·127-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{48} \cdot 5^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{48} \cdot 5^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7618819335\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7618819335\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 - T^{2} - 16 T^{3} - p T^{4} + p^{3} T^{6} \) |
| good | 3 | \( ( 1 - 2 T + 5 T^{2} - 8 T^{3} + 5 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 7 | \( 1 - 18 T^{2} + 239 T^{4} - 1868 T^{6} + 239 p^{2} T^{8} - 18 p^{4} T^{10} + p^{6} T^{12} \) |
| 11 | \( 1 - 2 p T^{2} + 311 T^{4} - 4116 T^{6} + 311 p^{2} T^{8} - 2 p^{5} T^{10} + p^{6} T^{12} \) |
| 13 | \( ( 1 + 8 T + 47 T^{2} + 192 T^{3} + 47 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 17 | \( 1 - 54 T^{2} + 1583 T^{4} - 31412 T^{6} + 1583 p^{2} T^{8} - 54 p^{4} T^{10} + p^{6} T^{12} \) |
| 19 | \( 1 - 2 p T^{2} + 1351 T^{4} - 26804 T^{6} + 1351 p^{2} T^{8} - 2 p^{5} T^{10} + p^{6} T^{12} \) |
| 23 | \( 1 - 98 T^{2} + 4335 T^{4} - 120044 T^{6} + 4335 p^{2} T^{8} - 98 p^{4} T^{10} + p^{6} T^{12} \) |
| 29 | \( ( 1 - 54 T^{2} + p^{2} T^{4} )^{3} \) |
| 31 | \( ( 1 + 8 T + 61 T^{2} + 368 T^{3} + 61 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 37 | \( ( 1 + 8 T + 79 T^{2} + 320 T^{3} + 79 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 41 | \( ( 1 + 2 T + 71 T^{2} - 20 T^{3} + 71 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 43 | \( ( 1 + 14 T + 149 T^{2} + 1104 T^{3} + 149 p T^{4} + 14 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 47 | \( 1 - 210 T^{2} + 21311 T^{4} - 1269644 T^{6} + 21311 p^{2} T^{8} - 210 p^{4} T^{10} + p^{6} T^{12} \) |
| 53 | \( ( 1 - 16 T + 231 T^{2} - 1776 T^{3} + 231 p T^{4} - 16 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 59 | \( 1 - 278 T^{2} + 35991 T^{4} - 2711444 T^{6} + 35991 p^{2} T^{8} - 278 p^{4} T^{10} + p^{6} T^{12} \) |
| 61 | \( 1 - 210 T^{2} + 18695 T^{4} - 1171868 T^{6} + 18695 p^{2} T^{8} - 210 p^{4} T^{10} + p^{6} T^{12} \) |
| 67 | \( ( 1 + 10 T + 141 T^{2} + 736 T^{3} + 141 p T^{4} + 10 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 71 | \( ( 1 + 12 T + 197 T^{2} + 1384 T^{3} + 197 p T^{4} + 12 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 73 | \( 1 - 38 T^{2} + 10495 T^{4} - 503444 T^{6} + 10495 p^{2} T^{8} - 38 p^{4} T^{10} + p^{6} T^{12} \) |
| 79 | \( ( 1 - 16 T + 269 T^{2} - 2400 T^{3} + 269 p T^{4} - 16 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 83 | \( ( 1 - 30 T + 509 T^{2} - 5504 T^{3} + 509 p T^{4} - 30 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 89 | \( ( 1 + 10 T + 151 T^{2} + 684 T^{3} + 151 p T^{4} + 10 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 97 | \( 1 - 278 T^{2} + 46735 T^{4} - 5290868 T^{6} + 46735 p^{2} T^{8} - 278 p^{4} T^{10} + p^{6} T^{12} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{12} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−5.06620802440505815920321241113, −4.93789088633096568732711380786, −4.74332271611027227039615818339, −4.73055137303972126295226225128, −4.57668564658521570333875182252, −4.19589291258586748997681064795, −3.97358549050841972780529778909, −3.86218921974129525415162486658, −3.72573475660776696736090782175, −3.51503324169879166391030694637, −3.29834097433060606514036264393, −3.18105372357371054406860186489, −3.12795050311415123732220028360, −3.06768673043374900177942004343, −2.46420850472633657284426807785, −2.44835801983859669160235899421, −2.43916895105153310372668383980, −2.29177299524063194949029651244, −2.04607977285097026106622138702, −1.96114210419891364673790608769, −1.71520714497584986542801945078, −1.32967067982207145138627129610, −0.960970335086214551497487309086, −0.30876343097875368341812398357, −0.18320740375440590333560405971,
0.18320740375440590333560405971, 0.30876343097875368341812398357, 0.960970335086214551497487309086, 1.32967067982207145138627129610, 1.71520714497584986542801945078, 1.96114210419891364673790608769, 2.04607977285097026106622138702, 2.29177299524063194949029651244, 2.43916895105153310372668383980, 2.44835801983859669160235899421, 2.46420850472633657284426807785, 3.06768673043374900177942004343, 3.12795050311415123732220028360, 3.18105372357371054406860186489, 3.29834097433060606514036264393, 3.51503324169879166391030694637, 3.72573475660776696736090782175, 3.86218921974129525415162486658, 3.97358549050841972780529778909, 4.19589291258586748997681064795, 4.57668564658521570333875182252, 4.73055137303972126295226225128, 4.74332271611027227039615818339, 4.93789088633096568732711380786, 5.06620802440505815920321241113
Plot not available for L-functions of degree greater than 10.
