| 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
−5.13077341445910684473441202879, −5.12467033400550934584875063219, −4.96275293020163801368425196248, −4.47388609697318311892120893156, −4.43935328883006265899204889788, −4.36704722236192362734842318888, −4.11055146910283247759856693277, −3.99017679348804031028670187841, −3.81601470312076775856144597963, −3.65647529477294157265530110032, −3.60998479776990115329557161954, −3.33981540516744409999881298230, −3.12487279012325326648858676213, −2.94210692351814796099076475427, −2.66505672829299754297055220035, −2.54583074090754424477561868137, −2.33184866609757018041268796582, −2.23324828343637446267091712207, −1.57525610902437685051853742601, −1.45294914689524267233933843123, −1.43488101746258903555871217936, −1.07546422311426779508781069030, −0.998787871345519049870340245082, −0.47622620208589242078484007047, −0.22594354613682051967255643017,
0.22594354613682051967255643017, 0.47622620208589242078484007047, 0.998787871345519049870340245082, 1.07546422311426779508781069030, 1.43488101746258903555871217936, 1.45294914689524267233933843123, 1.57525610902437685051853742601, 2.23324828343637446267091712207, 2.33184866609757018041268796582, 2.54583074090754424477561868137, 2.66505672829299754297055220035, 2.94210692351814796099076475427, 3.12487279012325326648858676213, 3.33981540516744409999881298230, 3.60998479776990115329557161954, 3.65647529477294157265530110032, 3.81601470312076775856144597963, 3.99017679348804031028670187841, 4.11055146910283247759856693277, 4.36704722236192362734842318888, 4.43935328883006265899204889788, 4.47388609697318311892120893156, 4.96275293020163801368425196248, 5.12467033400550934584875063219, 5.13077341445910684473441202879
Plot not available for L-functions of degree greater than 10.
