L(s) = 1 | + 3·2-s + 34·4-s + 258·7-s + 75·8-s + 402·11-s + 924·13-s + 774·14-s + 251·16-s + 276·17-s − 510·19-s + 1.20e3·22-s + 6.90e3·23-s + 4.84e3·25-s + 2.77e3·26-s + 8.77e3·28-s − 1.08e3·29-s + 6.41e3·31-s + 2.34e3·32-s + 828·34-s − 1.52e4·37-s − 1.53e3·38-s − 8.61e3·41-s + 5.83e4·43-s + 1.36e4·44-s + 2.07e4·46-s − 1.50e4·47-s + 1.15e3·49-s + ⋯ |
L(s) = 1 | + 0.530·2-s + 1.06·4-s + 1.99·7-s + 0.414·8-s + 1.00·11-s + 1.51·13-s + 1.05·14-s + 0.245·16-s + 0.231·17-s − 0.324·19-s + 0.531·22-s + 2.71·23-s + 1.54·25-s + 0.804·26-s + 2.11·28-s − 0.238·29-s + 1.19·31-s + 0.404·32-s + 0.122·34-s − 1.83·37-s − 0.171·38-s − 0.800·41-s + 4.81·43-s + 1.06·44-s + 1.44·46-s − 0.994·47-s + 0.0687·49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{16} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(6-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{16} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+5/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
\(L(3)\) |
\(\approx\) |
\(22.91427271\) |
\(L(\frac12)\) |
\(\approx\) |
\(22.91427271\) |
\(L(\frac{7}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 7 | \( 1 - 258 T + 9344 p T^{2} - 251256 p^{2} T^{3} + 4790655 p^{3} T^{4} - 251256 p^{7} T^{5} + 9344 p^{11} T^{6} - 258 p^{15} T^{7} + p^{20} T^{8} \) |
good | 2 | \( 1 - 3 T - 25 T^{2} + 51 p T^{3} + 259 p T^{4} - 1185 p^{2} T^{5} + 14739 p^{2} T^{6} - 1833 p^{3} T^{7} - 116575 p^{4} T^{8} - 1833 p^{8} T^{9} + 14739 p^{12} T^{10} - 1185 p^{17} T^{11} + 259 p^{21} T^{12} + 51 p^{26} T^{13} - 25 p^{30} T^{14} - 3 p^{35} T^{15} + p^{40} T^{16} \) |
| 5 | \( 1 - 4843 T^{2} - 24216 p^{2} T^{3} + 15905393 T^{4} + 96476544 p^{2} T^{5} + 149656086942 T^{6} - 275087633352 p^{2} T^{7} - 553760753377714 T^{8} - 275087633352 p^{7} T^{9} + 149656086942 p^{10} T^{10} + 96476544 p^{17} T^{11} + 15905393 p^{20} T^{12} - 24216 p^{27} T^{13} - 4843 p^{30} T^{14} + p^{40} T^{16} \) |
| 11 | \( 1 - 402 T - 244237 T^{2} + 23685786 T^{3} + 49988093357 T^{4} + 825774680136 p T^{5} - 7585190415099126 T^{6} - 907765850007682500 T^{7} + \)\(95\!\cdots\!06\)\( T^{8} - 907765850007682500 p^{5} T^{9} - 7585190415099126 p^{10} T^{10} + 825774680136 p^{16} T^{11} + 49988093357 p^{20} T^{12} + 23685786 p^{25} T^{13} - 244237 p^{30} T^{14} - 402 p^{35} T^{15} + p^{40} T^{16} \) |
| 13 | \( ( 1 - 462 T + 336749 T^{2} + 500754 T^{3} + 123849672672 T^{4} + 500754 p^{5} T^{5} + 336749 p^{10} T^{6} - 462 p^{15} T^{7} + p^{20} T^{8} )^{2} \) |
| 17 | \( 1 - 276 T - 4008500 T^{2} - 565843800 T^{3} + 9089867656090 T^{4} + 2604033925498692 T^{5} - 14267240611484004272 T^{6} - \)\(19\!\cdots\!84\)\( T^{7} + \)\(19\!\cdots\!07\)\( T^{8} - \)\(19\!\cdots\!84\)\( p^{5} T^{9} - 14267240611484004272 p^{10} T^{10} + 2604033925498692 p^{15} T^{11} + 9089867656090 p^{20} T^{12} - 565843800 p^{25} T^{13} - 4008500 p^{30} T^{14} - 276 p^{35} T^{15} + p^{40} T^{16} \) |
| 19 | \( 1 + 510 T - 3747425 T^{2} - 3937377630 T^{3} - 440016487199 T^{4} + 4115411338726080 T^{5} - 5704973911539330050 T^{6} + \)\(57\!\cdots\!00\)\( T^{7} + \)\(73\!\cdots\!50\)\( T^{8} + \)\(57\!\cdots\!00\)\( p^{5} T^{9} - 5704973911539330050 p^{10} T^{10} + 4115411338726080 p^{15} T^{11} - 440016487199 p^{20} T^{12} - 3937377630 p^{25} T^{13} - 3747425 p^{30} T^{14} + 510 p^{35} T^{15} + p^{40} T^{16} \) |
| 23 | \( 1 - 300 p T + 14742772 T^{2} + 5434854360 T^{3} - 64144356593702 T^{4} + 173260413144408420 T^{5} - \)\(23\!\cdots\!64\)\( T^{6} - \)\(19\!\cdots\!40\)\( T^{7} + \)\(99\!\cdots\!79\)\( T^{8} - \)\(19\!\cdots\!40\)\( p^{5} T^{9} - \)\(23\!\cdots\!64\)\( p^{10} T^{10} + 173260413144408420 p^{15} T^{11} - 64144356593702 p^{20} T^{12} + 5434854360 p^{25} T^{13} + 14742772 p^{30} T^{14} - 300 p^{36} T^{15} + p^{40} T^{16} \) |
| 29 | \( ( 1 + 540 T + 29394199 T^{2} - 29299685892 T^{3} + 772430149366772 T^{4} - 29299685892 p^{5} T^{5} + 29394199 p^{10} T^{6} + 540 p^{15} T^{7} + p^{20} T^{8} )^{2} \) |
| 31 | \( 1 - 6410 T - 55714636 T^{2} + 363334551632 T^{3} + 2227342403022119 T^{4} - 10547841526395999412 T^{5} - \)\(75\!\cdots\!84\)\( T^{6} + \)\(10\!\cdots\!10\)\( T^{7} + \)\(25\!\cdots\!76\)\( T^{8} + \)\(10\!\cdots\!10\)\( p^{5} T^{9} - \)\(75\!\cdots\!84\)\( p^{10} T^{10} - 10547841526395999412 p^{15} T^{11} + 2227342403022119 p^{20} T^{12} + 363334551632 p^{25} T^{13} - 55714636 p^{30} T^{14} - 6410 p^{35} T^{15} + p^{40} T^{16} \) |
| 37 | \( 1 + 15250 T + 2042879 T^{2} + 164587155110 T^{3} + 277810073119073 p T^{4} - 25563809512056184540 T^{5} - \)\(73\!\cdots\!82\)\( T^{6} - \)\(25\!\cdots\!80\)\( T^{7} - \)\(18\!\cdots\!10\)\( T^{8} - \)\(25\!\cdots\!80\)\( p^{5} T^{9} - \)\(73\!\cdots\!82\)\( p^{10} T^{10} - 25563809512056184540 p^{15} T^{11} + 277810073119073 p^{21} T^{12} + 164587155110 p^{25} T^{13} + 2042879 p^{30} T^{14} + 15250 p^{35} T^{15} + p^{40} T^{16} \) |
| 41 | \( ( 1 + 4308 T + 270683560 T^{2} + 2598901038204 T^{3} + 34018560333944366 T^{4} + 2598901038204 p^{5} T^{5} + 270683560 p^{10} T^{6} + 4308 p^{15} T^{7} + p^{20} T^{8} )^{2} \) |
| 43 | \( ( 1 - 29198 T + 787995265 T^{2} - 13076978932730 T^{3} + 187469286625894360 T^{4} - 13076978932730 p^{5} T^{5} + 787995265 p^{10} T^{6} - 29198 p^{15} T^{7} + p^{20} T^{8} )^{2} \) |
| 47 | \( 1 + 15060 T - 741226172 T^{2} - 6055210491384 T^{3} + 443152946335783210 T^{4} + \)\(21\!\cdots\!28\)\( T^{5} - \)\(15\!\cdots\!68\)\( T^{6} - \)\(13\!\cdots\!48\)\( T^{7} + \)\(43\!\cdots\!71\)\( T^{8} - \)\(13\!\cdots\!48\)\( p^{5} T^{9} - \)\(15\!\cdots\!68\)\( p^{10} T^{10} + \)\(21\!\cdots\!28\)\( p^{15} T^{11} + 443152946335783210 p^{20} T^{12} - 6055210491384 p^{25} T^{13} - 741226172 p^{30} T^{14} + 15060 p^{35} T^{15} + p^{40} T^{16} \) |
| 53 | \( 1 - 13692 T - 987403679 T^{2} + 2191455111540 T^{3} + 630826063270172605 T^{4} + \)\(25\!\cdots\!24\)\( T^{5} - \)\(29\!\cdots\!74\)\( T^{6} - \)\(54\!\cdots\!28\)\( T^{7} + \)\(11\!\cdots\!06\)\( T^{8} - \)\(54\!\cdots\!28\)\( p^{5} T^{9} - \)\(29\!\cdots\!74\)\( p^{10} T^{10} + \)\(25\!\cdots\!24\)\( p^{15} T^{11} + 630826063270172605 p^{20} T^{12} + 2191455111540 p^{25} T^{13} - 987403679 p^{30} T^{14} - 13692 p^{35} T^{15} + p^{40} T^{16} \) |
| 59 | \( 1 - 34830 T - 1834824737 T^{2} + 42321414019590 T^{3} + 3370913750880697081 T^{4} - \)\(47\!\cdots\!20\)\( T^{5} - \)\(33\!\cdots\!82\)\( T^{6} + \)\(90\!\cdots\!20\)\( T^{7} + \)\(30\!\cdots\!74\)\( T^{8} + \)\(90\!\cdots\!20\)\( p^{5} T^{9} - \)\(33\!\cdots\!82\)\( p^{10} T^{10} - \)\(47\!\cdots\!20\)\( p^{15} T^{11} + 3370913750880697081 p^{20} T^{12} + 42321414019590 p^{25} T^{13} - 1834824737 p^{30} T^{14} - 34830 p^{35} T^{15} + p^{40} T^{16} \) |
| 61 | \( 1 - 5364 T - 2817016532 T^{2} + 5158230942312 T^{3} + 4645398449535186778 T^{4} - \)\(12\!\cdots\!24\)\( T^{5} - \)\(55\!\cdots\!52\)\( T^{6} + \)\(26\!\cdots\!00\)\( T^{7} + \)\(52\!\cdots\!87\)\( T^{8} + \)\(26\!\cdots\!00\)\( p^{5} T^{9} - \)\(55\!\cdots\!52\)\( p^{10} T^{10} - \)\(12\!\cdots\!24\)\( p^{15} T^{11} + 4645398449535186778 p^{20} T^{12} + 5158230942312 p^{25} T^{13} - 2817016532 p^{30} T^{14} - 5364 p^{35} T^{15} + p^{40} T^{16} \) |
| 67 | \( 1 - 5994 T - 2518577501 T^{2} + 43127704637370 T^{3} + 2548139821971114109 T^{4} - \)\(68\!\cdots\!04\)\( T^{5} + \)\(21\!\cdots\!62\)\( T^{6} + \)\(47\!\cdots\!72\)\( T^{7} - \)\(17\!\cdots\!02\)\( T^{8} + \)\(47\!\cdots\!72\)\( p^{5} T^{9} + \)\(21\!\cdots\!62\)\( p^{10} T^{10} - \)\(68\!\cdots\!04\)\( p^{15} T^{11} + 2548139821971114109 p^{20} T^{12} + 43127704637370 p^{25} T^{13} - 2518577501 p^{30} T^{14} - 5994 p^{35} T^{15} + p^{40} T^{16} \) |
| 71 | \( ( 1 + 89268 T + 8738662172 T^{2} + 466802240277492 T^{3} + 25044546792022133910 T^{4} + 466802240277492 p^{5} T^{5} + 8738662172 p^{10} T^{6} + 89268 p^{15} T^{7} + p^{20} T^{8} )^{2} \) |
| 73 | \( 1 + 59638 T - 5098535509 T^{2} - 184967914738150 T^{3} + 26836978407770089433 T^{4} + \)\(52\!\cdots\!12\)\( T^{5} - \)\(81\!\cdots\!50\)\( T^{6} - \)\(30\!\cdots\!84\)\( T^{7} + \)\(20\!\cdots\!42\)\( T^{8} - \)\(30\!\cdots\!84\)\( p^{5} T^{9} - \)\(81\!\cdots\!50\)\( p^{10} T^{10} + \)\(52\!\cdots\!12\)\( p^{15} T^{11} + 26836978407770089433 p^{20} T^{12} - 184967914738150 p^{25} T^{13} - 5098535509 p^{30} T^{14} + 59638 p^{35} T^{15} + p^{40} T^{16} \) |
| 79 | \( 1 - 44062 T - 6584960844 T^{2} + 467510535225040 T^{3} + 20422039882069416887 T^{4} - \)\(19\!\cdots\!32\)\( T^{5} - \)\(53\!\cdots\!72\)\( T^{6} + \)\(30\!\cdots\!70\)\( T^{7} - \)\(76\!\cdots\!68\)\( T^{8} + \)\(30\!\cdots\!70\)\( p^{5} T^{9} - \)\(53\!\cdots\!72\)\( p^{10} T^{10} - \)\(19\!\cdots\!32\)\( p^{15} T^{11} + 20422039882069416887 p^{20} T^{12} + 467510535225040 p^{25} T^{13} - 6584960844 p^{30} T^{14} - 44062 p^{35} T^{15} + p^{40} T^{16} \) |
| 83 | \( ( 1 - 208446 T + 23363412401 T^{2} - 1724627286611514 T^{3} + \)\(11\!\cdots\!56\)\( T^{4} - 1724627286611514 p^{5} T^{5} + 23363412401 p^{10} T^{6} - 208446 p^{15} T^{7} + p^{20} T^{8} )^{2} \) |
| 89 | \( 1 + 77520 T + 479326112 T^{2} + 1563841943328288 T^{3} + \)\(12\!\cdots\!70\)\( T^{4} + \)\(11\!\cdots\!72\)\( T^{5} + \)\(11\!\cdots\!40\)\( T^{6} + \)\(89\!\cdots\!20\)\( T^{7} + \)\(66\!\cdots\!67\)\( T^{8} + \)\(89\!\cdots\!20\)\( p^{5} T^{9} + \)\(11\!\cdots\!40\)\( p^{10} T^{10} + \)\(11\!\cdots\!72\)\( p^{15} T^{11} + \)\(12\!\cdots\!70\)\( p^{20} T^{12} + 1563841943328288 p^{25} T^{13} + 479326112 p^{30} T^{14} + 77520 p^{35} T^{15} + p^{40} T^{16} \) |
| 97 | \( ( 1 + 188630 T + 43620869129 T^{2} + 4760435860011510 T^{3} + \)\(59\!\cdots\!64\)\( T^{4} + 4760435860011510 p^{5} T^{5} + 43620869129 p^{10} T^{6} + 188630 p^{15} T^{7} + p^{20} T^{8} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{16} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−5.96997733860352838886881321955, −5.89551399692766232195554371632, −5.58807260012321450387952585079, −5.51930212590954880588010106326, −5.04437905922265098265754137820, −5.02018787831344439120887918505, −4.94743286478410068285764423294, −4.60778866363151414139165423396, −4.55324412657194406911919212732, −4.22625444851309139243255449987, −4.07392881747717945016032973105, −3.83149634142669973452798908691, −3.51283288971617070275976505547, −3.25916490878785512029035979922, −3.09802947825728272826189930714, −2.92349361251216726866432611491, −2.44655853350267015015861280444, −2.43726799957064450052214165115, −2.09211284816022871516737471710, −1.60100842539679883761423261546, −1.38095645184881750988500101384, −1.28391182798183293440568151850, −0.972229520143510957034566323779, −0.900750391196867634479850628379, −0.28400935131352758421345348680,
0.28400935131352758421345348680, 0.900750391196867634479850628379, 0.972229520143510957034566323779, 1.28391182798183293440568151850, 1.38095645184881750988500101384, 1.60100842539679883761423261546, 2.09211284816022871516737471710, 2.43726799957064450052214165115, 2.44655853350267015015861280444, 2.92349361251216726866432611491, 3.09802947825728272826189930714, 3.25916490878785512029035979922, 3.51283288971617070275976505547, 3.83149634142669973452798908691, 4.07392881747717945016032973105, 4.22625444851309139243255449987, 4.55324412657194406911919212732, 4.60778866363151414139165423396, 4.94743286478410068285764423294, 5.02018787831344439120887918505, 5.04437905922265098265754137820, 5.51930212590954880588010106326, 5.58807260012321450387952585079, 5.89551399692766232195554371632, 5.96997733860352838886881321955
Plot not available for L-functions of degree greater than 10.