| L(s) = 1 | − 7·3-s + 3·5-s − 52·7-s + 26·9-s + 99·11-s − 21·15-s + 9·17-s + 143·19-s + 364·21-s + 15·23-s − 30·25-s − 73·27-s + 348·29-s − 205·31-s − 693·33-s − 156·35-s + 249·37-s + 78·45-s − 75·47-s + 1.70e3·49-s − 63·51-s + 645·53-s + 297·55-s − 1.00e3·57-s + 321·59-s + 1.70e3·61-s − 1.35e3·63-s + ⋯ |
| L(s) = 1 | − 1.34·3-s + 0.268·5-s − 2.80·7-s + 0.962·9-s + 2.71·11-s − 0.361·15-s + 0.128·17-s + 1.72·19-s + 3.78·21-s + 0.135·23-s − 0.239·25-s − 0.520·27-s + 2.22·29-s − 1.18·31-s − 3.65·33-s − 0.753·35-s + 1.10·37-s + 0.258·45-s − 0.232·47-s + 4.96·49-s − 0.172·51-s + 1.67·53-s + 0.728·55-s − 2.32·57-s + 0.708·59-s + 3.58·61-s − 2.70·63-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{36} \cdot 7^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{36} \cdot 7^{6}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(3.029144828\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.029144828\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 + 52 T + 143 p T^{2} + 2008 p T^{3} + 143 p^{4} T^{4} + 52 p^{6} T^{5} + p^{9} T^{6} \) |
| good | 3 | \( 1 + 7 T + 23 T^{2} + 52 T^{3} - 481 T^{4} - 3035 T^{5} - 10382 T^{6} - 3035 p^{3} T^{7} - 481 p^{6} T^{8} + 52 p^{9} T^{9} + 23 p^{12} T^{10} + 7 p^{15} T^{11} + p^{18} T^{12} \) |
| 5 | \( 1 - 3 T + 39 T^{2} - 108 T^{3} - 1743 T^{4} - 5721 T^{5} - 3079618 T^{6} - 5721 p^{3} T^{7} - 1743 p^{6} T^{8} - 108 p^{9} T^{9} + 39 p^{12} T^{10} - 3 p^{15} T^{11} + p^{18} T^{12} \) |
| 11 | \( 1 - 9 p T + 7983 T^{2} - 42444 p T^{3} + 24786639 T^{4} - 1085499369 T^{5} + 42926385890 T^{6} - 1085499369 p^{3} T^{7} + 24786639 p^{6} T^{8} - 42444 p^{10} T^{9} + 7983 p^{12} T^{10} - 9 p^{16} T^{11} + p^{18} T^{12} \) |
| 13 | \( 1 - 606 p T^{2} + 33176199 T^{4} - 88766411348 T^{6} + 33176199 p^{6} T^{8} - 606 p^{13} T^{10} + p^{18} T^{12} \) |
| 17 | \( 1 - 9 T + 99 T^{2} - 648 T^{3} - 221787 T^{4} + 214195761 T^{5} - 237206836690 T^{6} + 214195761 p^{3} T^{7} - 221787 p^{6} T^{8} - 648 p^{9} T^{9} + 99 p^{12} T^{10} - 9 p^{15} T^{11} + p^{18} T^{12} \) |
| 19 | \( 1 - 143 T - 5665 T^{2} + 708 p^{2} T^{3} + 217547935 T^{4} - 7596399877 T^{5} - 1014293563550 T^{6} - 7596399877 p^{3} T^{7} + 217547935 p^{6} T^{8} + 708 p^{11} T^{9} - 5665 p^{12} T^{10} - 143 p^{15} T^{11} + p^{18} T^{12} \) |
| 23 | \( 1 - 15 T + 25275 T^{2} - 378000 T^{3} + 329438955 T^{4} - 2856530865 T^{5} + 4126653853346 T^{6} - 2856530865 p^{3} T^{7} + 329438955 p^{6} T^{8} - 378000 p^{9} T^{9} + 25275 p^{12} T^{10} - 15 p^{15} T^{11} + p^{18} T^{12} \) |
| 29 | \( ( 1 - 6 p T + 45663 T^{2} - 3864972 T^{3} + 45663 p^{3} T^{4} - 6 p^{7} T^{5} + p^{9} T^{6} )^{2} \) |
| 31 | \( 1 + 205 T - 18277 T^{2} - 9001152 T^{3} - 349839533 T^{4} + 89508615347 T^{5} + 21347063732914 T^{6} + 89508615347 p^{3} T^{7} - 349839533 p^{6} T^{8} - 9001152 p^{9} T^{9} - 18277 p^{12} T^{10} + 205 p^{15} T^{11} + p^{18} T^{12} \) |
| 37 | \( 1 - 249 T - 56433 T^{2} + 13334072 T^{3} + 2695969773 T^{4} - 187262229591 T^{5} - 151167420301962 T^{6} - 187262229591 p^{3} T^{7} + 2695969773 p^{6} T^{8} + 13334072 p^{9} T^{9} - 56433 p^{12} T^{10} - 249 p^{15} T^{11} + p^{18} T^{12} \) |
| 41 | \( 1 - 4350 p T^{2} + 20420082255 T^{4} - 1736555393379940 T^{6} + 20420082255 p^{6} T^{8} - 4350 p^{13} T^{10} + p^{18} T^{12} \) |
| 43 | \( 1 - 225246 T^{2} + 31219429287 T^{4} - 3038313220434308 T^{6} + 31219429287 p^{6} T^{8} - 225246 p^{12} T^{10} + p^{18} T^{12} \) |
| 47 | \( 1 + 75 T - 271293 T^{2} - 10090392 T^{3} + 46775899635 T^{4} + 768612310653 T^{5} - 5545818129678302 T^{6} + 768612310653 p^{3} T^{7} + 46775899635 p^{6} T^{8} - 10090392 p^{9} T^{9} - 271293 p^{12} T^{10} + 75 p^{15} T^{11} + p^{18} T^{12} \) |
| 53 | \( 1 - 645 T + 85935 T^{2} + 48846432 T^{3} - 24599221275 T^{4} + 5896942109805 T^{5} - 1714098743019578 T^{6} + 5896942109805 p^{3} T^{7} - 24599221275 p^{6} T^{8} + 48846432 p^{9} T^{9} + 85935 p^{12} T^{10} - 645 p^{15} T^{11} + p^{18} T^{12} \) |
| 59 | \( 1 - 321 T - 543849 T^{2} + 57872532 T^{3} + 242992413159 T^{4} - 14127840402531 T^{5} - 55338821261641982 T^{6} - 14127840402531 p^{3} T^{7} + 242992413159 p^{6} T^{8} + 57872532 p^{9} T^{9} - 543849 p^{12} T^{10} - 321 p^{15} T^{11} + p^{18} T^{12} \) |
| 61 | \( 1 - 1707 T + 1959075 T^{2} - 1686160944 T^{3} + 1222226403561 T^{4} - 729202413995805 T^{5} + 376156083256198486 T^{6} - 729202413995805 p^{3} T^{7} + 1222226403561 p^{6} T^{8} - 1686160944 p^{9} T^{9} + 1959075 p^{12} T^{10} - 1707 p^{15} T^{11} + p^{18} T^{12} \) |
| 67 | \( 1 - 447 T + 454983 T^{2} - 173605860 T^{3} + 25906463103 T^{4} + 61606109318883 T^{5} - 14080566334108718 T^{6} + 61606109318883 p^{3} T^{7} + 25906463103 p^{6} T^{8} - 173605860 p^{9} T^{9} + 454983 p^{12} T^{10} - 447 p^{15} T^{11} + p^{18} T^{12} \) |
| 71 | \( 1 - 730518 T^{2} + 329494751679 T^{4} - 139370995079650996 T^{6} + 329494751679 p^{6} T^{8} - 730518 p^{12} T^{10} + p^{18} T^{12} \) |
| 73 | \( 1 - 705 T + 711327 T^{2} - 384684660 T^{3} + 124877058765 T^{4} + 15186280400661 T^{5} - 17616940504633370 T^{6} + 15186280400661 p^{3} T^{7} + 124877058765 p^{6} T^{8} - 384684660 p^{9} T^{9} + 711327 p^{12} T^{10} - 705 p^{15} T^{11} + p^{18} T^{12} \) |
| 79 | \( 1 - 3447 T + 6344259 T^{2} - 8216462232 T^{3} + 8250767428323 T^{4} - 6929574062947137 T^{5} + 5138476947017253250 T^{6} - 6929574062947137 p^{3} T^{7} + 8250767428323 p^{6} T^{8} - 8216462232 p^{9} T^{9} + 6344259 p^{12} T^{10} - 3447 p^{15} T^{11} + p^{18} T^{12} \) |
| 83 | \( ( 1 - 12 T + 858849 T^{2} + 294490104 T^{3} + 858849 p^{3} T^{4} - 12 p^{6} T^{5} + p^{9} T^{6} )^{2} \) |
| 89 | \( 1 + 2607 T + 4818303 T^{2} + 6655201740 T^{3} + 7793954699517 T^{4} + 7742714792403333 T^{5} + 6946049341042583942 T^{6} + 7742714792403333 p^{3} T^{7} + 7793954699517 p^{6} T^{8} + 6655201740 p^{9} T^{9} + 4818303 p^{12} T^{10} + 2607 p^{15} T^{11} + p^{18} T^{12} \) |
| 97 | \( 1 - 4437342 T^{2} + 9010629855423 T^{4} - 10556356459893838148 T^{6} + 9010629855423 p^{6} T^{8} - 4437342 p^{12} T^{10} + p^{18} T^{12} \) |
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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
−5.40995046895051254049890331506, −5.40078541106327339244213312001, −5.34403698766255430220178015540, −5.32634622613020891503001572069, −4.99178441908734743634873291268, −4.35991352752197713308002926062, −4.29525040655486407993916476009, −4.21673637566036278828424820041, −4.19032901831465597236115148955, −3.76955692542206893888037312599, −3.66843429162971075196147983656, −3.57386246963290306929087926173, −3.33167722111985286135958246334, −3.02225681362660782168733489126, −2.87679558991654803057034636555, −2.60977599736772476314857897331, −2.45983541118618373788721977817, −2.09200628111468260606438664574, −1.76438510042765424849900268721, −1.60970065482185188728057903331, −0.929313392384516992040831821713, −0.893323567492594651895904102317, −0.802018566437416210893770641774, −0.73819249365762733704126715891, −0.20900474593128644689055307226,
0.20900474593128644689055307226, 0.73819249365762733704126715891, 0.802018566437416210893770641774, 0.893323567492594651895904102317, 0.929313392384516992040831821713, 1.60970065482185188728057903331, 1.76438510042765424849900268721, 2.09200628111468260606438664574, 2.45983541118618373788721977817, 2.60977599736772476314857897331, 2.87679558991654803057034636555, 3.02225681362660782168733489126, 3.33167722111985286135958246334, 3.57386246963290306929087926173, 3.66843429162971075196147983656, 3.76955692542206893888037312599, 4.19032901831465597236115148955, 4.21673637566036278828424820041, 4.29525040655486407993916476009, 4.35991352752197713308002926062, 4.99178441908734743634873291268, 5.32634622613020891503001572069, 5.34403698766255430220178015540, 5.40078541106327339244213312001, 5.40995046895051254049890331506
Plot not available for L-functions of degree greater than 10.
