| L(s) = 1 | + 96·2-s + 5.37e3·4-s + 2.29e5·8-s − 2.27e4·9-s + 1.10e5·11-s + 8.25e6·16-s − 2.18e6·18-s + 1.06e7·22-s + 2.43e5·23-s + 2.70e6·25-s + 1.04e7·29-s + 2.64e8·32-s − 1.22e8·36-s + 2.43e7·37-s + 7.47e7·43-s + 5.94e8·44-s + 2.33e7·46-s + 2.59e8·50-s + 1.37e8·53-s + 1.00e9·58-s + 7.75e9·64-s − 2.02e8·67-s + 6.14e8·71-s − 5.22e9·72-s + 2.33e9·74-s − 1.82e8·79-s − 2.05e8·81-s + ⋯ |
| L(s) = 1 | + 4.24·2-s + 21/2·4-s + 19.7·8-s − 1.15·9-s + 2.27·11-s + 63/2·16-s − 4.90·18-s + 9.66·22-s + 0.181·23-s + 1.38·25-s + 2.74·29-s + 44.5·32-s − 12.1·36-s + 2.13·37-s + 3.33·43-s + 23.9·44-s + 0.769·46-s + 5.87·50-s + 2.39·53-s + 11.6·58-s + 57.7·64-s − 1.23·67-s + 2.86·71-s − 22.9·72-s + 9.05·74-s − 0.526·79-s − 0.531·81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{6} \cdot 7^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(10-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{6} \cdot 7^{12}\right)^{s/2} \, \Gamma_{\C}(s+9/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(506.6458479\) |
| \(L(\frac12)\) |
\(\approx\) |
\(506.6458479\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( ( 1 - p^{4} T )^{6} \) |
| 7 | \( 1 \) |
| good | 3 | \( 1 + 22774 T^{2} + 80505379 p^{2} T^{4} + 98674660588 p^{4} T^{6} + 80505379 p^{20} T^{8} + 22774 p^{36} T^{10} + p^{54} T^{12} \) |
| 5 | \( 1 - 2703348 T^{2} - 74739443193 p^{2} T^{4} + 27023141472180728 p^{4} T^{6} - 74739443193 p^{20} T^{8} - 2703348 p^{36} T^{10} + p^{54} T^{12} \) |
| 11 | \( ( 1 - 55332 T + 6363390333 T^{2} - 247868065093768 T^{3} + 6363390333 p^{9} T^{4} - 55332 p^{18} T^{5} + p^{27} T^{6} )^{2} \) |
| 13 | \( 1 + 50176523580 T^{2} + \)\(11\!\cdots\!99\)\( T^{4} + \)\(15\!\cdots\!48\)\( T^{6} + \)\(11\!\cdots\!99\)\( p^{18} T^{8} + 50176523580 p^{36} T^{10} + p^{54} T^{12} \) |
| 17 | \( 1 + 335677241280 T^{2} + \)\(48\!\cdots\!95\)\( T^{4} + \)\(52\!\cdots\!64\)\( T^{6} + \)\(48\!\cdots\!95\)\( p^{18} T^{8} + 335677241280 p^{36} T^{10} + p^{54} T^{12} \) |
| 19 | \( 1 + 1215201325782 T^{2} + \)\(74\!\cdots\!43\)\( T^{4} + \)\(29\!\cdots\!48\)\( T^{6} + \)\(74\!\cdots\!43\)\( p^{18} T^{8} + 1215201325782 p^{36} T^{10} + p^{54} T^{12} \) |
| 23 | \( ( 1 - 121692 T + 2333259771765 T^{2} - 1393456861115377352 T^{3} + 2333259771765 p^{9} T^{4} - 121692 p^{18} T^{5} + p^{27} T^{6} )^{2} \) |
| 29 | \( ( 1 - 5231820 T + 39716097937755 T^{2} - \)\(13\!\cdots\!84\)\( T^{3} + 39716097937755 p^{9} T^{4} - 5231820 p^{18} T^{5} + p^{27} T^{6} )^{2} \) |
| 31 | \( 1 + 42158956552170 T^{2} + \)\(14\!\cdots\!33\)\( p T^{4} - \)\(97\!\cdots\!60\)\( T^{6} + \)\(14\!\cdots\!33\)\( p^{19} T^{8} + 42158956552170 p^{36} T^{10} + p^{54} T^{12} \) |
| 37 | \( ( 1 - 12166284 T + 264123307533651 T^{2} - \)\(34\!\cdots\!36\)\( T^{3} + 264123307533651 p^{9} T^{4} - 12166284 p^{18} T^{5} + p^{27} T^{6} )^{2} \) |
| 41 | \( 1 + 1578191530625088 T^{2} + \)\(11\!\cdots\!39\)\( T^{4} + \)\(47\!\cdots\!56\)\( T^{6} + \)\(11\!\cdots\!39\)\( p^{18} T^{8} + 1578191530625088 p^{36} T^{10} + p^{54} T^{12} \) |
| 43 | \( ( 1 - 37379292 T + 977188199141085 T^{2} - \)\(16\!\cdots\!64\)\( T^{3} + 977188199141085 p^{9} T^{4} - 37379292 p^{18} T^{5} + p^{27} T^{6} )^{2} \) |
| 47 | \( 1 + 831234330438186 T^{2} + \)\(69\!\cdots\!11\)\( T^{4} + \)\(19\!\cdots\!92\)\( T^{6} + \)\(69\!\cdots\!11\)\( p^{18} T^{8} + 831234330438186 p^{36} T^{10} + p^{54} T^{12} \) |
| 53 | \( ( 1 - 68907654 T + 9369785854095771 T^{2} - \)\(38\!\cdots\!96\)\( T^{3} + 9369785854095771 p^{9} T^{4} - 68907654 p^{18} T^{5} + p^{27} T^{6} )^{2} \) |
| 59 | \( 1 + 27647673784317702 T^{2} + \)\(36\!\cdots\!79\)\( T^{4} + \)\(33\!\cdots\!24\)\( T^{6} + \)\(36\!\cdots\!79\)\( p^{18} T^{8} + 27647673784317702 p^{36} T^{10} + p^{54} T^{12} \) |
| 61 | \( 1 + 23878734814251180 T^{2} + \)\(44\!\cdots\!55\)\( T^{4} + \)\(65\!\cdots\!92\)\( T^{6} + \)\(44\!\cdots\!55\)\( p^{18} T^{8} + 23878734814251180 p^{36} T^{10} + p^{54} T^{12} \) |
| 67 | \( ( 1 + 101471688 T + 68926735612618617 T^{2} + \)\(47\!\cdots\!60\)\( T^{3} + 68926735612618617 p^{9} T^{4} + 101471688 p^{18} T^{5} + p^{27} T^{6} )^{2} \) |
| 71 | \( ( 1 - 307146384 T + 96942609915077445 T^{2} - \)\(15\!\cdots\!60\)\( T^{3} + 96942609915077445 p^{9} T^{4} - 307146384 p^{18} T^{5} + p^{27} T^{6} )^{2} \) |
| 73 | \( 1 + 201982863234329376 T^{2} + \)\(22\!\cdots\!47\)\( T^{4} + \)\(16\!\cdots\!96\)\( T^{6} + \)\(22\!\cdots\!47\)\( p^{18} T^{8} + 201982863234329376 p^{36} T^{10} + p^{54} T^{12} \) |
| 79 | \( ( 1 + 91185048 T + 303663600453471453 T^{2} + \)\(18\!\cdots\!04\)\( T^{3} + 303663600453471453 p^{9} T^{4} + 91185048 p^{18} T^{5} + p^{27} T^{6} )^{2} \) |
| 83 | \( 1 + 516413512013081334 T^{2} + \)\(18\!\cdots\!79\)\( T^{4} + \)\(39\!\cdots\!64\)\( T^{6} + \)\(18\!\cdots\!79\)\( p^{18} T^{8} + 516413512013081334 p^{36} T^{10} + p^{54} T^{12} \) |
| 89 | \( 1 + 617696186000737056 T^{2} + \)\(31\!\cdots\!87\)\( T^{4} + \)\(15\!\cdots\!52\)\( T^{6} + \)\(31\!\cdots\!87\)\( p^{18} T^{8} + 617696186000737056 p^{36} T^{10} + p^{54} T^{12} \) |
| 97 | \( 1 + 3878015556873763488 T^{2} + \)\(66\!\cdots\!15\)\( T^{4} + \)\(65\!\cdots\!00\)\( T^{6} + \)\(66\!\cdots\!15\)\( p^{18} T^{8} + 3878015556873763488 p^{36} T^{10} + p^{54} T^{12} \) |
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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
−5.97163360984667814995557953300, −5.50937104220546687446307884920, −5.42300471885029579783915776005, −5.17283568765116529169750626615, −5.08060710596409427109593682215, −4.89193573226115636496663598108, −4.46400217660326999443210425749, −4.33229535132290875725857916693, −4.14992697805289216129774698866, −4.11747606565458466373594273028, −3.69934151805079656393346246716, −3.57093698265328527601562295000, −3.55787023065481210204875494218, −2.95173666694056566230911881303, −2.69859775179073521113182404677, −2.68005509307630478600459374424, −2.49951460899808714829704082299, −2.45584177975435826284747684588, −2.09005357542198063736768763969, −1.47229380512198269865278908179, −1.26275221593491605643673849207, −1.17203388790207479149506172589, −1.00953427447194184390644519582, −0.68102625300608395360508820719, −0.36809561925808697777940461695,
0.36809561925808697777940461695, 0.68102625300608395360508820719, 1.00953427447194184390644519582, 1.17203388790207479149506172589, 1.26275221593491605643673849207, 1.47229380512198269865278908179, 2.09005357542198063736768763969, 2.45584177975435826284747684588, 2.49951460899808714829704082299, 2.68005509307630478600459374424, 2.69859775179073521113182404677, 2.95173666694056566230911881303, 3.55787023065481210204875494218, 3.57093698265328527601562295000, 3.69934151805079656393346246716, 4.11747606565458466373594273028, 4.14992697805289216129774698866, 4.33229535132290875725857916693, 4.46400217660326999443210425749, 4.89193573226115636496663598108, 5.08060710596409427109593682215, 5.17283568765116529169750626615, 5.42300471885029579783915776005, 5.50937104220546687446307884920, 5.97163360984667814995557953300
Plot not available for L-functions of degree greater than 10.
