| L(s) = 1 | + 31·4-s + 264·13-s + 405·16-s − 2.17e3·25-s + 6.39e3·37-s − 2.05e3·49-s + 8.18e3·52-s − 9.73e3·61-s + 3.63e3·64-s − 3.09e4·73-s + 8.54e4·97-s − 6.73e4·100-s − 8.96e4·109-s + 1.63e5·121-s + 127-s + 131-s + 137-s + 139-s + 1.98e5·148-s + 149-s + 151-s + 157-s + 163-s + 167-s − 9.41e4·169-s + 173-s + 179-s + ⋯ |
| L(s) = 1 | + 1.93·4-s + 1.56·13-s + 1.58·16-s − 3.47·25-s + 4.66·37-s − 6/7·49-s + 3.02·52-s − 2.61·61-s + 0.888·64-s − 5.80·73-s + 9.07·97-s − 6.73·100-s − 7.54·109-s + 11.1·121-s + 6.20e−5·127-s + 5.82e−5·131-s + 5.32e−5·137-s + 5.17e−5·139-s + 9.04·148-s + 4.50e−5·149-s + 4.38e−5·151-s + 4.05e−5·157-s + 3.76e−5·163-s + 3.58e−5·167-s − 3.29·169-s + 3.34e−5·173-s + 3.12e−5·179-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{24} \cdot 7^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(5-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{24} \cdot 7^{12}\right)^{s/2} \, \Gamma_{\C}(s+2)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(2.918183421\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.918183421\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - 31 T^{2} + 139 p^{2} T^{4} - 65 p^{7} T^{6} + 139 p^{10} T^{8} - 31 p^{16} T^{10} + p^{24} T^{12} \) |
| 3 | \( 1 \) |
| 7 | \( ( 1 + p^{3} T^{2} )^{6} \) |
| good | 5 | \( ( 1 + 1086 T^{2} + 65107 p T^{4} - 71263996 T^{6} + 65107 p^{9} T^{8} + 1086 p^{16} T^{10} + p^{24} T^{12} )^{2} \) |
| 11 | \( ( 1 - 81934 T^{2} + 2878176607 T^{4} - 55428162243556 T^{6} + 2878176607 p^{8} T^{8} - 81934 p^{16} T^{10} + p^{24} T^{12} )^{2} \) |
| 13 | \( ( 1 - 66 T + 34415 T^{2} + 31172 T^{3} + 34415 p^{4} T^{4} - 66 p^{8} T^{5} + p^{12} T^{6} )^{4} \) |
| 17 | \( ( 1 + 437278 T^{2} + 84627922687 T^{4} + 9192168369842116 T^{6} + 84627922687 p^{8} T^{8} + 437278 p^{16} T^{10} + p^{24} T^{12} )^{2} \) |
| 19 | \( ( 1 - 358790 T^{2} + 37438864495 T^{4} - 1997489304021140 T^{6} + 37438864495 p^{8} T^{8} - 358790 p^{16} T^{10} + p^{24} T^{12} )^{2} \) |
| 23 | \( ( 1 - 207854 T^{2} + 103719656575 T^{4} - 43641444041795492 T^{6} + 103719656575 p^{8} T^{8} - 207854 p^{16} T^{10} + p^{24} T^{12} )^{2} \) |
| 29 | \( ( 1 + 2666694 T^{2} + 3653667212271 T^{4} + 3206170012152266260 T^{6} + 3653667212271 p^{8} T^{8} + 2666694 p^{16} T^{10} + p^{24} T^{12} )^{2} \) |
| 31 | \( ( 1 - 2924134 T^{2} + 4732733742991 T^{4} - 5244431426352028756 T^{6} + 4732733742991 p^{8} T^{8} - 2924134 p^{16} T^{10} + p^{24} T^{12} )^{2} \) |
| 37 | \( ( 1 - 1598 T + 5085647 T^{2} - 4772206020 T^{3} + 5085647 p^{4} T^{4} - 1598 p^{8} T^{5} + p^{12} T^{6} )^{4} \) |
| 41 | \( ( 1 + 6782430 T^{2} + 31258648671551 T^{4} + \)\(10\!\cdots\!80\)\( T^{6} + 31258648671551 p^{8} T^{8} + 6782430 p^{16} T^{10} + p^{24} T^{12} )^{2} \) |
| 43 | \( ( 1 - 7678502 T^{2} + 40148020461487 T^{4} - \)\(17\!\cdots\!24\)\( T^{6} + 40148020461487 p^{8} T^{8} - 7678502 p^{16} T^{10} + p^{24} T^{12} )^{2} \) |
| 47 | \( ( 1 - 17766950 T^{2} + 152216917088527 T^{4} - \)\(86\!\cdots\!84\)\( T^{6} + 152216917088527 p^{8} T^{8} - 17766950 p^{16} T^{10} + p^{24} T^{12} )^{2} \) |
| 53 | \( ( 1 + 1903142 T^{2} + 124006524294319 T^{4} + \)\(29\!\cdots\!20\)\( T^{6} + 124006524294319 p^{8} T^{8} + 1903142 p^{16} T^{10} + p^{24} T^{12} )^{2} \) |
| 59 | \( ( 1 - 42993478 T^{2} + 961005248271791 T^{4} - \)\(14\!\cdots\!40\)\( T^{6} + 961005248271791 p^{8} T^{8} - 42993478 p^{16} T^{10} + p^{24} T^{12} )^{2} \) |
| 61 | \( ( 1 + 2434 T + 36296623 T^{2} + 990182764 p T^{3} + 36296623 p^{4} T^{4} + 2434 p^{8} T^{5} + p^{12} T^{6} )^{4} \) |
| 67 | \( ( 1 - 97685094 T^{2} + 4346739635938415 T^{4} - \)\(11\!\cdots\!96\)\( T^{6} + 4346739635938415 p^{8} T^{8} - 97685094 p^{16} T^{10} + p^{24} T^{12} )^{2} \) |
| 71 | \( ( 1 + 24130514 T^{2} + 97449779259391 T^{4} - \)\(11\!\cdots\!24\)\( T^{6} + 97449779259391 p^{8} T^{8} + 24130514 p^{16} T^{10} + p^{24} T^{12} )^{2} \) |
| 73 | \( ( 1 + 7734 T + 72961775 T^{2} + 334144466228 T^{3} + 72961775 p^{4} T^{4} + 7734 p^{8} T^{5} + p^{12} T^{6} )^{4} \) |
| 79 | \( ( 1 - 99683526 T^{2} + 6446186586103695 T^{4} - \)\(29\!\cdots\!48\)\( T^{6} + 6446186586103695 p^{8} T^{8} - 99683526 p^{16} T^{10} + p^{24} T^{12} )^{2} \) |
| 83 | \( ( 1 + 42599194 T^{2} - 1228838124870929 T^{4} - \)\(20\!\cdots\!00\)\( T^{6} - 1228838124870929 p^{8} T^{8} + 42599194 p^{16} T^{10} + p^{24} T^{12} )^{2} \) |
| 89 | \( ( 1 + 215516638 T^{2} + 25953249330452927 T^{4} + \)\(19\!\cdots\!96\)\( T^{6} + 25953249330452927 p^{8} T^{8} + 215516638 p^{16} T^{10} + p^{24} T^{12} )^{2} \) |
| 97 | \( ( 1 - 21354 T + 335284239 T^{2} - 3633666267148 T^{3} + 335284239 p^{4} T^{4} - 21354 p^{8} T^{5} + p^{12} T^{6} )^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{24} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−3.30483084703131335602679316314, −3.25873450231111457770563249070, −3.08217645079709401347260344697, −3.00091948064176420668359312601, −2.99640778069558007802740032658, −2.72742546192955524241273293369, −2.58995091570370205689571013393, −2.44357596197156460010389712884, −2.32752793302834015244664921785, −2.27270628141146791120067510565, −2.20156763182495597986551216578, −1.99823100836563747014821471897, −1.92052980612824984506233853810, −1.84611326141067083836592551309, −1.56126219932815021614611363505, −1.48790868472464006765031415079, −1.28636840848782234623564252518, −1.20784485495639062063710314845, −1.04408581965589496670256998440, −1.00488440746664386131202737304, −0.894576150193770883454586828822, −0.58113643969235180967996217510, −0.31155645702234118370904301030, −0.21661664607744613495891064093, −0.090479669816355414931630635582,
0.090479669816355414931630635582, 0.21661664607744613495891064093, 0.31155645702234118370904301030, 0.58113643969235180967996217510, 0.894576150193770883454586828822, 1.00488440746664386131202737304, 1.04408581965589496670256998440, 1.20784485495639062063710314845, 1.28636840848782234623564252518, 1.48790868472464006765031415079, 1.56126219932815021614611363505, 1.84611326141067083836592551309, 1.92052980612824984506233853810, 1.99823100836563747014821471897, 2.20156763182495597986551216578, 2.27270628141146791120067510565, 2.32752793302834015244664921785, 2.44357596197156460010389712884, 2.58995091570370205689571013393, 2.72742546192955524241273293369, 2.99640778069558007802740032658, 3.00091948064176420668359312601, 3.08217645079709401347260344697, 3.25873450231111457770563249070, 3.30483084703131335602679316314
Plot not available for L-functions of degree greater than 10.
