| L(s) = 1 | + 94·4-s + 112·7-s − 246·9-s + 4.06e3·16-s − 1.48e4·25-s + 1.05e4·28-s − 2.31e4·36-s + 2.74e4·37-s − 7.38e4·43-s + 9.75e3·49-s − 2.75e4·63-s + 9.82e4·64-s + 7.93e4·67-s − 2.47e5·79-s − 3.19e4·81-s − 1.39e6·100-s + 1.05e6·109-s + 4.55e5·112-s + 8.98e5·121-s + 127-s + 131-s + 137-s + 139-s − 1.00e6·144-s + 2.58e6·148-s + 149-s + 151-s + ⋯ |
| L(s) = 1 | + 2.93·4-s + 0.863·7-s − 1.01·9-s + 3.97·16-s − 4.75·25-s + 2.53·28-s − 2.97·36-s + 3.29·37-s − 6.09·43-s + 0.580·49-s − 0.874·63-s + 2.99·64-s + 2.15·67-s − 4.45·79-s − 0.541·81-s − 13.9·100-s + 8.47·109-s + 3.43·112-s + 5.58·121-s + 5.50e−6·127-s + 5.09e−6·131-s + 4.55e−6·137-s + 4.38e−6·139-s − 4.02·144-s + 9.68·148-s + 3.69e−6·149-s + 3.56e−6·151-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{12} \cdot 7^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(6-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{12} \cdot 7^{12}\right)^{s/2} \, \Gamma_{\C}(s+5/2)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(6.812477741\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.812477741\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + 82 p T^{2} + 3425 p^{3} T^{4} + 9188 p^{7} T^{6} + 3425 p^{13} T^{8} + 82 p^{21} T^{10} + p^{30} T^{12} \) |
| 7 | \( ( 1 - 8 p T - 25 p T^{2} + 8944 p^{3} T^{3} - 25 p^{6} T^{4} - 8 p^{11} T^{5} + p^{15} T^{6} )^{2} \) |
| good | 2 | \( ( 1 - 47 T^{2} + 5 p^{8} T^{4} - 1373 p^{4} T^{6} + 5 p^{18} T^{8} - 47 p^{20} T^{10} + p^{30} T^{12} )^{2} \) |
| 5 | \( ( 1 + 7422 T^{2} + 45037803 T^{4} + 153732902524 T^{6} + 45037803 p^{10} T^{8} + 7422 p^{20} T^{10} + p^{30} T^{12} )^{2} \) |
| 11 | \( ( 1 - 449462 T^{2} + 113070860327 T^{4} - 19709117341993844 T^{6} + 113070860327 p^{10} T^{8} - 449462 p^{20} T^{10} + p^{30} T^{12} )^{2} \) |
| 13 | \( ( 1 - 816402 T^{2} + 113284257339 T^{4} + 39146525353939228 T^{6} + 113284257339 p^{10} T^{8} - 816402 p^{20} T^{10} + p^{30} T^{12} )^{2} \) |
| 17 | \( ( 1 + 4555650 T^{2} + 12132254659551 T^{4} + 20340216096880619644 T^{6} + 12132254659551 p^{10} T^{8} + 4555650 p^{20} T^{10} + p^{30} T^{12} )^{2} \) |
| 19 | \( ( 1 - 10123818 T^{2} + 45169989998811 T^{4} - \)\(12\!\cdots\!52\)\( T^{6} + 45169989998811 p^{10} T^{8} - 10123818 p^{20} T^{10} + p^{30} T^{12} )^{2} \) |
| 23 | \( ( 1 - 35418710 T^{2} + 542203157102207 T^{4} - \)\(45\!\cdots\!84\)\( T^{6} + 542203157102207 p^{10} T^{8} - 35418710 p^{20} T^{10} + p^{30} T^{12} )^{2} \) |
| 29 | \( ( 1 - 119981462 T^{2} + 6059091136763351 T^{4} - \)\(56\!\cdots\!76\)\( p T^{6} + 6059091136763351 p^{10} T^{8} - 119981462 p^{20} T^{10} + p^{30} T^{12} )^{2} \) |
| 31 | \( ( 1 - 138042750 T^{2} + 8600352415587903 T^{4} - \)\(31\!\cdots\!00\)\( T^{6} + 8600352415587903 p^{10} T^{8} - 138042750 p^{20} T^{10} + p^{30} T^{12} )^{2} \) |
| 37 | \( ( 1 - 6866 T + 122124203 T^{2} - 548854640588 T^{3} + 122124203 p^{5} T^{4} - 6866 p^{10} T^{5} + p^{15} T^{6} )^{4} \) |
| 41 | \( ( 1 + 359708658 T^{2} + 63862412350351791 T^{4} + \)\(80\!\cdots\!72\)\( T^{6} + 63862412350351791 p^{10} T^{8} + 359708658 p^{20} T^{10} + p^{30} T^{12} )^{2} \) |
| 43 | \( ( 1 + 18460 T + 443121953 T^{2} + 4802973736744 T^{3} + 443121953 p^{5} T^{4} + 18460 p^{10} T^{5} + p^{15} T^{6} )^{4} \) |
| 47 | \( ( 1 + 212120106 T^{2} + 155469989261639919 T^{4} + \)\(20\!\cdots\!88\)\( T^{6} + 155469989261639919 p^{10} T^{8} + 212120106 p^{20} T^{10} + p^{30} T^{12} )^{2} \) |
| 53 | \( ( 1 - 1151960102 T^{2} + 783650903988257639 T^{4} - \)\(37\!\cdots\!72\)\( T^{6} + 783650903988257639 p^{10} T^{8} - 1151960102 p^{20} T^{10} + p^{30} T^{12} )^{2} \) |
| 59 | \( ( 1 + 951086886 T^{2} + 1554382177197754539 T^{4} + \)\(93\!\cdots\!04\)\( T^{6} + 1554382177197754539 p^{10} T^{8} + 951086886 p^{20} T^{10} + p^{30} T^{12} )^{2} \) |
| 61 | \( ( 1 - 3377279442 T^{2} + 5458813899270420891 T^{4} - \)\(55\!\cdots\!28\)\( T^{6} + 5458813899270420891 p^{10} T^{8} - 3377279442 p^{20} T^{10} + p^{30} T^{12} )^{2} \) |
| 67 | \( ( 1 - 19836 T + 3115193913 T^{2} - 38905579256168 T^{3} + 3115193913 p^{5} T^{4} - 19836 p^{10} T^{5} + p^{15} T^{6} )^{4} \) |
| 71 | \( ( 1 - 5943889790 T^{2} + 19424775775472096975 T^{4} - \)\(42\!\cdots\!00\)\( T^{6} + 19424775775472096975 p^{10} T^{8} - 5943889790 p^{20} T^{10} + p^{30} T^{12} )^{2} \) |
| 73 | \( ( 1 - 5910429174 T^{2} + 16372224800949034239 T^{4} - \)\(34\!\cdots\!64\)\( T^{6} + 16372224800949034239 p^{10} T^{8} - 5910429174 p^{20} T^{10} + p^{30} T^{12} )^{2} \) |
| 79 | \( ( 1 + 61776 T + 8765607177 T^{2} + 326859059630752 T^{3} + 8765607177 p^{5} T^{4} + 61776 p^{10} T^{5} + p^{15} T^{6} )^{4} \) |
| 83 | \( ( 1 + 10222546902 T^{2} + 31285392100211416539 T^{4} + \)\(49\!\cdots\!72\)\( T^{6} + 31285392100211416539 p^{10} T^{8} + 10222546902 p^{20} T^{10} + p^{30} T^{12} )^{2} \) |
| 89 | \( ( 1 + 16201315650 T^{2} + \)\(17\!\cdots\!23\)\( T^{4} + \)\(11\!\cdots\!48\)\( T^{6} + \)\(17\!\cdots\!23\)\( p^{10} T^{8} + 16201315650 p^{20} T^{10} + p^{30} T^{12} )^{2} \) |
| 97 | \( ( 1 - 31367046486 T^{2} + \)\(51\!\cdots\!75\)\( T^{4} - \)\(53\!\cdots\!96\)\( T^{6} + \)\(51\!\cdots\!75\)\( p^{10} T^{8} - 31367046486 p^{20} T^{10} + p^{30} T^{12} )^{2} \) |
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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
−5.92634868520121605833532746188, −5.91508432046575067653423981011, −5.85408042800250985365412143575, −5.66818453022616795506663652808, −5.62150791721412454698424784164, −5.39831175283800354483346163180, −4.96788068259591657949180281872, −4.88855566014675110696910846684, −4.56457490570615532020311725680, −4.53618401521729189447121888764, −4.31905631515313288723733751327, −3.82694600976095372123911756961, −3.79465271022214912122947768145, −3.53722294388708886316723898460, −3.23636324193975020888997313609, −3.14837008572207202202926695387, −2.74570393790464471227984520129, −2.55823905972264835698155094103, −2.17950203234029764118238098247, −2.14782822407711496181097665206, −1.74774827895559304788128698119, −1.61138893560570523559857649571, −1.47111560695398160288067341002, −0.55632725601983360977821292890, −0.32346105444179861885746270660,
0.32346105444179861885746270660, 0.55632725601983360977821292890, 1.47111560695398160288067341002, 1.61138893560570523559857649571, 1.74774827895559304788128698119, 2.14782822407711496181097665206, 2.17950203234029764118238098247, 2.55823905972264835698155094103, 2.74570393790464471227984520129, 3.14837008572207202202926695387, 3.23636324193975020888997313609, 3.53722294388708886316723898460, 3.79465271022214912122947768145, 3.82694600976095372123911756961, 4.31905631515313288723733751327, 4.53618401521729189447121888764, 4.56457490570615532020311725680, 4.88855566014675110696910846684, 4.96788068259591657949180281872, 5.39831175283800354483346163180, 5.62150791721412454698424784164, 5.66818453022616795506663652808, 5.85408042800250985365412143575, 5.91508432046575067653423981011, 5.92634868520121605833532746188
Plot not available for L-functions of degree greater than 10.
