| L(s) = 1 | + 151·4-s + 4.11e3·9-s + 5.71e3·11-s − 1.73e4·16-s + 1.80e5·19-s − 2.11e5·29-s − 1.42e5·31-s + 6.20e5·36-s − 5.16e5·41-s + 8.63e5·44-s − 3.52e5·49-s − 2.18e6·59-s − 5.25e6·61-s − 4.70e6·64-s − 1.59e7·71-s + 2.72e7·76-s − 1.00e7·79-s + 1.39e7·81-s − 3.10e7·89-s + 2.35e7·99-s − 4.18e7·101-s + 8.18e6·109-s − 3.19e7·116-s − 7.17e7·121-s − 2.15e7·124-s + 127-s + 131-s + ⋯ |
| L(s) = 1 | + 1.17·4-s + 1.88·9-s + 1.29·11-s − 1.05·16-s + 6.04·19-s − 1.60·29-s − 0.861·31-s + 2.21·36-s − 1.16·41-s + 1.52·44-s − 3/7·49-s − 1.38·59-s − 2.96·61-s − 2.24·64-s − 5.28·71-s + 7.13·76-s − 2.30·79-s + 2.91·81-s − 4.66·89-s + 2.43·99-s − 4.04·101-s + 0.605·109-s − 1.89·116-s − 3.68·121-s − 1.01·124-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{12} \cdot 7^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(8-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{12} \cdot 7^{6}\right)^{s/2} \, \Gamma_{\C}(s+7/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(7.767880225\) |
| \(L(\frac12)\) |
\(\approx\) |
\(7.767880225\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 7 | \( ( 1 + p^{6} T^{2} )^{3} \) |
| good | 2 | \( 1 - 151 T^{2} + 5021 p^{3} T^{4} - 248663 p^{4} T^{6} + 5021 p^{17} T^{8} - 151 p^{28} T^{10} + p^{42} T^{12} \) |
| 3 | \( 1 - 4112 T^{2} + 327920 p^{2} T^{4} + 68368954 p^{4} T^{6} + 327920 p^{16} T^{8} - 4112 p^{28} T^{10} + p^{42} T^{12} \) |
| 11 | \( ( 1 - 2858 T + 48121346 T^{2} - 81807946328 T^{3} + 48121346 p^{7} T^{4} - 2858 p^{14} T^{5} + p^{21} T^{6} )^{2} \) |
| 13 | \( 1 - 201510104 T^{2} + 17704287937950216 T^{4} - \)\(11\!\cdots\!02\)\( T^{6} + 17704287937950216 p^{14} T^{8} - 201510104 p^{28} T^{10} + p^{42} T^{12} \) |
| 17 | \( 1 - 206979408 T^{2} + 116598141343535088 T^{4} - \)\(10\!\cdots\!54\)\( T^{6} + 116598141343535088 p^{14} T^{8} - 206979408 p^{28} T^{10} + p^{42} T^{12} \) |
| 19 | \( ( 1 - 4756 p T + 4989916869 T^{2} - 178920526659912 T^{3} + 4989916869 p^{7} T^{4} - 4756 p^{15} T^{5} + p^{21} T^{6} )^{2} \) |
| 23 | \( 1 - 14769589314 T^{2} + \)\(10\!\cdots\!83\)\( T^{4} - \)\(44\!\cdots\!12\)\( T^{6} + \)\(10\!\cdots\!83\)\( p^{14} T^{8} - 14769589314 p^{28} T^{10} + p^{42} T^{12} \) |
| 29 | \( ( 1 + 105644 T + 45411470804 T^{2} + 3626050792485942 T^{3} + 45411470804 p^{7} T^{4} + 105644 p^{14} T^{5} + p^{21} T^{6} )^{2} \) |
| 31 | \( ( 1 + 71408 T + 75837702973 T^{2} + 3792566456498976 T^{3} + 75837702973 p^{7} T^{4} + 71408 p^{14} T^{5} + p^{21} T^{6} )^{2} \) |
| 37 | \( 1 + 205032286 T^{2} + \)\(24\!\cdots\!91\)\( T^{4} - \)\(15\!\cdots\!92\)\( T^{6} + \)\(24\!\cdots\!91\)\( p^{14} T^{8} + 205032286 p^{28} T^{10} + p^{42} T^{12} \) |
| 41 | \( ( 1 + 258086 T + 518669568571 T^{2} + 89774522487987884 T^{3} + 518669568571 p^{7} T^{4} + 258086 p^{14} T^{5} + p^{21} T^{6} )^{2} \) |
| 43 | \( 1 - 1497008728282 T^{2} + \)\(96\!\cdots\!43\)\( T^{4} - \)\(34\!\cdots\!76\)\( T^{6} + \)\(96\!\cdots\!43\)\( p^{14} T^{8} - 1497008728282 p^{28} T^{10} + p^{42} T^{12} \) |
| 47 | \( 1 - 2472952376216 T^{2} + \)\(27\!\cdots\!96\)\( T^{4} - \)\(17\!\cdots\!58\)\( T^{6} + \)\(27\!\cdots\!96\)\( p^{14} T^{8} - 2472952376216 p^{28} T^{10} + p^{42} T^{12} \) |
| 53 | \( 1 - 53370084418 p T^{2} + \)\(64\!\cdots\!03\)\( T^{4} - \)\(82\!\cdots\!72\)\( T^{6} + \)\(64\!\cdots\!03\)\( p^{14} T^{8} - 53370084418 p^{29} T^{10} + p^{42} T^{12} \) |
| 59 | \( ( 1 + 1091968 T + 3461847231673 T^{2} + 572519879612090624 T^{3} + 3461847231673 p^{7} T^{4} + 1091968 p^{14} T^{5} + p^{21} T^{6} )^{2} \) |
| 61 | \( ( 1 + 2628822 T + 1478898635151 T^{2} - 4035874228726439428 T^{3} + 1478898635151 p^{7} T^{4} + 2628822 p^{14} T^{5} + p^{21} T^{6} )^{2} \) |
| 67 | \( 1 - 30625296895506 T^{2} + \)\(41\!\cdots\!63\)\( T^{4} - \)\(32\!\cdots\!28\)\( T^{6} + \)\(41\!\cdots\!63\)\( p^{14} T^{8} - 30625296895506 p^{28} T^{10} + p^{42} T^{12} \) |
| 71 | \( ( 1 + 7975232 T + 47843354463269 T^{2} + \)\(16\!\cdots\!96\)\( T^{3} + 47843354463269 p^{7} T^{4} + 7975232 p^{14} T^{5} + p^{21} T^{6} )^{2} \) |
| 73 | \( 1 - 1500380884106 T^{2} + \)\(23\!\cdots\!39\)\( T^{4} - \)\(54\!\cdots\!28\)\( T^{6} + \)\(23\!\cdots\!39\)\( p^{14} T^{8} - 1500380884106 p^{28} T^{10} + p^{42} T^{12} \) |
| 79 | \( ( 1 + 5040874 T + 36626740795278 T^{2} + \)\(13\!\cdots\!32\)\( T^{3} + 36626740795278 p^{7} T^{4} + 5040874 p^{14} T^{5} + p^{21} T^{6} )^{2} \) |
| 83 | \( 1 - 65592305423810 T^{2} + \)\(33\!\cdots\!87\)\( T^{4} - \)\(10\!\cdots\!80\)\( T^{6} + \)\(33\!\cdots\!87\)\( p^{14} T^{8} - 65592305423810 p^{28} T^{10} + p^{42} T^{12} \) |
| 89 | \( ( 1 + 15507394 T + 199440114124459 T^{2} + \)\(14\!\cdots\!12\)\( T^{3} + 199440114124459 p^{7} T^{4} + 15507394 p^{14} T^{5} + p^{21} T^{6} )^{2} \) |
| 97 | \( 1 - 142509217688640 T^{2} + \)\(15\!\cdots\!32\)\( T^{4} - \)\(15\!\cdots\!70\)\( T^{6} + \)\(15\!\cdots\!32\)\( p^{14} T^{8} - 142509217688640 p^{28} T^{10} + p^{42} T^{12} \) |
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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
−5.65441210067645717574648565271, −5.51582950859431899711539170703, −5.33360211061857871754280723211, −5.16283019906980469059383814003, −4.63476323795894363503866724969, −4.62130997556442376260786614583, −4.52925810954720287700922675537, −4.09499897292906990396393274774, −4.07811802261551552065994012006, −3.78628964911670400199606306326, −3.54247637051127886467027276425, −3.10447044278271762143470120638, −3.00265185108153997673505624907, −2.98921538166808238351889082436, −2.78831447982532202009473851205, −2.63291111960716870218092757189, −1.75904533514421264967389733882, −1.71290019236310366752463786204, −1.52092069001802737502964837270, −1.50473526634227009231683854276, −1.49708910643810293619507462584, −1.18746648251861861998606282441, −0.59783469030687751242913994037, −0.54417219535857303000044773303, −0.15892957045383610772891797494,
0.15892957045383610772891797494, 0.54417219535857303000044773303, 0.59783469030687751242913994037, 1.18746648251861861998606282441, 1.49708910643810293619507462584, 1.50473526634227009231683854276, 1.52092069001802737502964837270, 1.71290019236310366752463786204, 1.75904533514421264967389733882, 2.63291111960716870218092757189, 2.78831447982532202009473851205, 2.98921538166808238351889082436, 3.00265185108153997673505624907, 3.10447044278271762143470120638, 3.54247637051127886467027276425, 3.78628964911670400199606306326, 4.07811802261551552065994012006, 4.09499897292906990396393274774, 4.52925810954720287700922675537, 4.62130997556442376260786614583, 4.63476323795894363503866724969, 5.16283019906980469059383814003, 5.33360211061857871754280723211, 5.51582950859431899711539170703, 5.65441210067645717574648565271
Plot not available for L-functions of degree greater than 10.
