| L(s) = 1 | − 6·2-s − 9·3-s + 12·4-s + 54·6-s + 14·7-s + 16·8-s + 27·9-s − 33·11-s − 108·12-s − 84·13-s − 84·14-s − 144·16-s + 75·17-s − 162·18-s + 43·19-s − 126·21-s + 198·22-s + 199·23-s − 144·24-s + 367·25-s + 504·26-s + 54·27-s + 168·28-s − 682·29-s − 74·31-s + 288·32-s + 297·33-s + ⋯ |
| L(s) = 1 | − 2.12·2-s − 1.73·3-s + 3/2·4-s + 3.67·6-s + 0.755·7-s + 0.707·8-s + 9-s − 0.904·11-s − 2.59·12-s − 1.79·13-s − 1.60·14-s − 9/4·16-s + 1.07·17-s − 2.12·18-s + 0.519·19-s − 1.30·21-s + 1.91·22-s + 1.80·23-s − 1.22·24-s + 2.93·25-s + 3.80·26-s + 0.384·27-s + 1.13·28-s − 4.36·29-s − 0.428·31-s + 1.59·32-s + 1.56·33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{6} \cdot 3^{6} \cdot 7^{6} \cdot 11^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{6} \cdot 3^{6} \cdot 7^{6} \cdot 11^{6}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.1548027203\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1548027203\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( ( 1 + p T + p^{2} T^{2} )^{3} \) |
| 3 | \( ( 1 + p T + p^{2} T^{2} )^{3} \) |
| 7 | \( 1 - 2 p T + 5 p^{2} T^{2} - 32 p^{3} T^{3} + 5 p^{5} T^{4} - 2 p^{7} T^{5} + p^{9} T^{6} \) |
| 11 | \( ( 1 + p T + p^{2} T^{2} )^{3} \) |
| good | 5 | \( 1 - 367 T^{2} - 8 T^{3} + 88814 T^{4} + 1468 T^{5} - 12929859 T^{6} + 1468 p^{3} T^{7} + 88814 p^{6} T^{8} - 8 p^{9} T^{9} - 367 p^{12} T^{10} + p^{18} T^{12} \) |
| 13 | \( ( 1 + 42 T + 5827 T^{2} + 177152 T^{3} + 5827 p^{3} T^{4} + 42 p^{6} T^{5} + p^{9} T^{6} )^{2} \) |
| 17 | \( 1 - 75 T - 7363 T^{2} + 304474 T^{3} + 56931575 T^{4} - 31874743 p T^{5} - 318675055734 T^{6} - 31874743 p^{4} T^{7} + 56931575 p^{6} T^{8} + 304474 p^{9} T^{9} - 7363 p^{12} T^{10} - 75 p^{15} T^{11} + p^{18} T^{12} \) |
| 19 | \( 1 - 43 T - 18115 T^{2} + 271790 T^{3} + 237328017 T^{4} - 1528803991 T^{5} - 1855982623034 T^{6} - 1528803991 p^{3} T^{7} + 237328017 p^{6} T^{8} + 271790 p^{9} T^{9} - 18115 p^{12} T^{10} - 43 p^{15} T^{11} + p^{18} T^{12} \) |
| 23 | \( 1 - 199 T - 5599 T^{2} + 894910 T^{3} + 509065455 T^{4} - 34087124711 T^{5} - 2666140626338 T^{6} - 34087124711 p^{3} T^{7} + 509065455 p^{6} T^{8} + 894910 p^{9} T^{9} - 5599 p^{12} T^{10} - 199 p^{15} T^{11} + p^{18} T^{12} \) |
| 29 | \( ( 1 + 341 T + 96946 T^{2} + 16807039 T^{3} + 96946 p^{3} T^{4} + 341 p^{6} T^{5} + p^{9} T^{6} )^{2} \) |
| 31 | \( 1 + 74 T - 31177 T^{2} + 1611138 T^{3} + 210769078 T^{4} - 75749448434 T^{5} + 8299932908131 T^{6} - 75749448434 p^{3} T^{7} + 210769078 p^{6} T^{8} + 1611138 p^{9} T^{9} - 31177 p^{12} T^{10} + 74 p^{15} T^{11} + p^{18} T^{12} \) |
| 37 | \( 1 - 381 T - 45313 T^{2} + 6888440 T^{3} + 11495319177 T^{4} - 1193145187123 T^{5} - 371793709486834 T^{6} - 1193145187123 p^{3} T^{7} + 11495319177 p^{6} T^{8} + 6888440 p^{9} T^{9} - 45313 p^{12} T^{10} - 381 p^{15} T^{11} + p^{18} T^{12} \) |
| 41 | \( ( 1 + 12 T + 51559 T^{2} - 21600692 T^{3} + 51559 p^{3} T^{4} + 12 p^{6} T^{5} + p^{9} T^{6} )^{2} \) |
| 43 | \( ( 1 + 257 T + 66106 T^{2} + 33762609 T^{3} + 66106 p^{3} T^{4} + 257 p^{6} T^{5} + p^{9} T^{6} )^{2} \) |
| 47 | \( 1 - 327 T - 92331 T^{2} + 78413930 T^{3} - 4457237205 T^{4} - 4299074570163 T^{5} + 2116631191516718 T^{6} - 4299074570163 p^{3} T^{7} - 4457237205 p^{6} T^{8} + 78413930 p^{9} T^{9} - 92331 p^{12} T^{10} - 327 p^{15} T^{11} + p^{18} T^{12} \) |
| 53 | \( 1 - 392 T - 302191 T^{2} + 41832624 T^{3} + 96714010526 T^{4} - 5745577379588 T^{5} - 15585414833593651 T^{6} - 5745577379588 p^{3} T^{7} + 96714010526 p^{6} T^{8} + 41832624 p^{9} T^{9} - 302191 p^{12} T^{10} - 392 p^{15} T^{11} + p^{18} T^{12} \) |
| 59 | \( 1 - 631 T - 210931 T^{2} + 55494018 T^{3} + 107131656011 T^{4} - 2695190966275 T^{5} - 27299993869939954 T^{6} - 2695190966275 p^{3} T^{7} + 107131656011 p^{6} T^{8} + 55494018 p^{9} T^{9} - 210931 p^{12} T^{10} - 631 p^{15} T^{11} + p^{18} T^{12} \) |
| 61 | \( 1 - 128 T - 68931 T^{2} + 384998248 T^{3} - 37425023142 T^{4} - 18526093924828 T^{5} + 60079945414190589 T^{6} - 18526093924828 p^{3} T^{7} - 37425023142 p^{6} T^{8} + 384998248 p^{9} T^{9} - 68931 p^{12} T^{10} - 128 p^{15} T^{11} + p^{18} T^{12} \) |
| 67 | \( 1 - 676 T + 318459 T^{2} - 52633212 T^{3} - 36333275822 T^{4} - 7664375259832 T^{5} + 7725369450471043 T^{6} - 7664375259832 p^{3} T^{7} - 36333275822 p^{6} T^{8} - 52633212 p^{9} T^{9} + 318459 p^{12} T^{10} - 676 p^{15} T^{11} + p^{18} T^{12} \) |
| 71 | \( ( 1 + 585 T + 740824 T^{2} + 419448829 T^{3} + 740824 p^{3} T^{4} + 585 p^{6} T^{5} + p^{9} T^{6} )^{2} \) |
| 73 | \( 1 - 714 T + 147965 T^{2} - 73661974 T^{3} - 75769411086 T^{4} + 59537927843930 T^{5} - 1809995553771355 T^{6} + 59537927843930 p^{3} T^{7} - 75769411086 p^{6} T^{8} - 73661974 p^{9} T^{9} + 147965 p^{12} T^{10} - 714 p^{15} T^{11} + p^{18} T^{12} \) |
| 79 | \( 1 - 1110 T + 494835 T^{2} + 102795834 T^{3} - 331184509230 T^{4} + 197574300873510 T^{5} - 115667054129921173 T^{6} + 197574300873510 p^{3} T^{7} - 331184509230 p^{6} T^{8} + 102795834 p^{9} T^{9} + 494835 p^{12} T^{10} - 1110 p^{15} T^{11} + p^{18} T^{12} \) |
| 83 | \( ( 1 - 524 T + 524753 T^{2} + 35631112 T^{3} + 524753 p^{3} T^{4} - 524 p^{6} T^{5} + p^{9} T^{6} )^{2} \) |
| 89 | \( 1 + 184 T - 1328091 T^{2} + 336629576 T^{3} + 902684098342 T^{4} - 320468489589608 T^{5} - 621909381654234223 T^{6} - 320468489589608 p^{3} T^{7} + 902684098342 p^{6} T^{8} + 336629576 p^{9} T^{9} - 1328091 p^{12} T^{10} + 184 p^{15} T^{11} + p^{18} T^{12} \) |
| 97 | \( ( 1 + 583 T + 238770 T^{2} + 600520875 T^{3} + 238770 p^{3} T^{4} + 583 p^{6} T^{5} + p^{9} T^{6} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{12} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−5.36507998532578524906547684851, −5.35816660840933256282488710346, −5.29408507342247912783812553543, −5.05844460324373116418954226549, −4.92100014359279468725851589816, −4.73828536745781875656117380904, −4.52737105493478588786732403545, −4.42374328992255652772497366520, −4.03415289499673534876683117008, −3.85558474683889859093817321100, −3.63649066936247311381462314707, −3.35554125020789833878288977583, −3.13719790742853581708982351230, −2.90998248499199512096756180743, −2.81608062788974752998399047093, −2.26477435638923593463618259713, −2.16169415849927982297582220845, −2.01635331402896023873861851531, −1.82365857881294948782814895417, −1.17928547656236770088435221647, −1.08397836289890119789703949921, −0.901180547121830242219514095761, −0.820917604932115872114868272855, −0.27954064021565612961070707297, −0.16507352222930982363145909018,
0.16507352222930982363145909018, 0.27954064021565612961070707297, 0.820917604932115872114868272855, 0.901180547121830242219514095761, 1.08397836289890119789703949921, 1.17928547656236770088435221647, 1.82365857881294948782814895417, 2.01635331402896023873861851531, 2.16169415849927982297582220845, 2.26477435638923593463618259713, 2.81608062788974752998399047093, 2.90998248499199512096756180743, 3.13719790742853581708982351230, 3.35554125020789833878288977583, 3.63649066936247311381462314707, 3.85558474683889859093817321100, 4.03415289499673534876683117008, 4.42374328992255652772497366520, 4.52737105493478588786732403545, 4.73828536745781875656117380904, 4.92100014359279468725851589816, 5.05844460324373116418954226549, 5.29408507342247912783812553543, 5.35816660840933256282488710346, 5.36507998532578524906547684851
Plot not available for L-functions of degree greater than 10.
