| L(s) = 1 | − 24·7-s + 6.61e3·9-s + 1.07e3·17-s − 8.66e4·23-s + 2.74e5·25-s + 3.13e5·31-s + 1.81e6·41-s + 1.70e6·47-s − 4.10e6·49-s − 1.58e5·63-s − 1.98e7·71-s + 1.15e7·73-s + 2.00e6·79-s + 1.87e7·81-s + 9.66e6·89-s − 1.05e7·97-s − 3.53e7·103-s + 2.56e7·113-s − 2.58e4·119-s + 3.53e7·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 7.11e6·153-s + ⋯ |
| L(s) = 1 | − 0.0264·7-s + 3.02·9-s + 0.0531·17-s − 1.48·23-s + 3.51·25-s + 1.89·31-s + 4.12·41-s + 2.39·47-s − 4.98·49-s − 0.0799·63-s − 6.59·71-s + 3.48·73-s + 0.458·79-s + 3.91·81-s + 1.45·89-s − 1.16·97-s − 3.18·103-s + 1.66·113-s − 0.00140·119-s + 1.81·121-s + 0.160·153-s + 0.0392·161-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{48}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(8-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{48}\right)^{s/2} \, \Gamma_{\C}(s+7/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(18.21552020\) |
| \(L(\frac12)\) |
\(\approx\) |
\(18.21552020\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| good | 3 | \( 1 - 6614 T^{2} + 2781887 p^{2} T^{4} - 828078932 p^{4} T^{6} + 2781887 p^{16} T^{8} - 6614 p^{28} T^{10} + p^{42} T^{12} \) |
| 5 | \( 1 - 274978 T^{2} + 40067152359 T^{4} - 149715918285404 p^{2} T^{6} + 40067152359 p^{14} T^{8} - 274978 p^{28} T^{10} + p^{42} T^{12} \) |
| 7 | \( ( 1 + 12 T + 293379 p T^{2} - 53820568 T^{3} + 293379 p^{8} T^{4} + 12 p^{14} T^{5} + p^{21} T^{6} )^{2} \) |
| 11 | \( 1 - 35331942 T^{2} + 909287095694631 T^{4} - \)\(19\!\cdots\!96\)\( T^{6} + 909287095694631 p^{14} T^{8} - 35331942 p^{28} T^{10} + p^{42} T^{12} \) |
| 13 | \( 1 - 177163570 T^{2} + 19168281772450679 T^{4} - \)\(14\!\cdots\!28\)\( T^{6} + 19168281772450679 p^{14} T^{8} - 177163570 p^{28} T^{10} + p^{42} T^{12} \) |
| 17 | \( ( 1 - 538 T + 599769055 T^{2} - 6045136516972 T^{3} + 599769055 p^{7} T^{4} - 538 p^{14} T^{5} + p^{21} T^{6} )^{2} \) |
| 19 | \( 1 - 2815712758 T^{2} + 4640981944627350551 T^{4} - \)\(49\!\cdots\!44\)\( T^{6} + 4640981944627350551 p^{14} T^{8} - 2815712758 p^{28} T^{10} + p^{42} T^{12} \) |
| 23 | \( ( 1 + 43316 T + 7071007589 T^{2} + 331921661265816 T^{3} + 7071007589 p^{7} T^{4} + 43316 p^{14} T^{5} + p^{21} T^{6} )^{2} \) |
| 29 | \( 1 - 37523550962 T^{2} + \)\(24\!\cdots\!91\)\( T^{4} + \)\(41\!\cdots\!92\)\( T^{6} + \)\(24\!\cdots\!91\)\( p^{14} T^{8} - 37523550962 p^{28} T^{10} + p^{42} T^{12} \) |
| 31 | \( ( 1 - 156928 T + 39853704861 T^{2} - 10335750019347968 T^{3} + 39853704861 p^{7} T^{4} - 156928 p^{14} T^{5} + p^{21} T^{6} )^{2} \) |
| 37 | \( 1 - 231379937346 T^{2} + \)\(32\!\cdots\!27\)\( T^{4} - \)\(33\!\cdots\!32\)\( T^{6} + \)\(32\!\cdots\!27\)\( p^{14} T^{8} - 231379937346 p^{28} T^{10} + p^{42} T^{12} \) |
| 41 | \( ( 1 - 909698 T + 288771195831 T^{2} - 42998200165964540 T^{3} + 288771195831 p^{7} T^{4} - 909698 p^{14} T^{5} + p^{21} T^{6} )^{2} \) |
| 43 | \( 1 - 1168158143558 T^{2} + \)\(61\!\cdots\!83\)\( T^{4} - \)\(20\!\cdots\!64\)\( T^{6} + \)\(61\!\cdots\!83\)\( p^{14} T^{8} - 1168158143558 p^{28} T^{10} + p^{42} T^{12} \) |
| 47 | \( ( 1 - 852952 T + 725351339437 T^{2} - 444762288735514960 T^{3} + 725351339437 p^{7} T^{4} - 852952 p^{14} T^{5} + p^{21} T^{6} )^{2} \) |
| 53 | \( 1 - 2509555980834 T^{2} + \)\(50\!\cdots\!15\)\( T^{4} - \)\(62\!\cdots\!24\)\( T^{6} + \)\(50\!\cdots\!15\)\( p^{14} T^{8} - 2509555980834 p^{28} T^{10} + p^{42} T^{12} \) |
| 59 | \( 1 - 10842792880518 T^{2} + \)\(52\!\cdots\!71\)\( T^{4} - \)\(15\!\cdots\!24\)\( T^{6} + \)\(52\!\cdots\!71\)\( p^{14} T^{8} - 10842792880518 p^{28} T^{10} + p^{42} T^{12} \) |
| 61 | \( 1 - 9799407007890 T^{2} + \)\(48\!\cdots\!35\)\( T^{4} - \)\(17\!\cdots\!28\)\( T^{6} + \)\(48\!\cdots\!35\)\( p^{14} T^{8} - 9799407007890 p^{28} T^{10} + p^{42} T^{12} \) |
| 67 | \( 1 - 11373989589558 T^{2} + \)\(10\!\cdots\!95\)\( T^{4} - \)\(75\!\cdots\!76\)\( T^{6} + \)\(10\!\cdots\!95\)\( p^{14} T^{8} - 11373989589558 p^{28} T^{10} + p^{42} T^{12} \) |
| 71 | \( ( 1 + 9941468 T + 59931979536597 T^{2} + \)\(21\!\cdots\!16\)\( T^{3} + 59931979536597 p^{7} T^{4} + 9941468 p^{14} T^{5} + p^{21} T^{6} )^{2} \) |
| 73 | \( ( 1 - 5794470 T + 37828195636791 T^{2} - \)\(12\!\cdots\!88\)\( T^{3} + 37828195636791 p^{7} T^{4} - 5794470 p^{14} T^{5} + p^{21} T^{6} )^{2} \) |
| 79 | \( ( 1 - 1004504 T + 48614168717389 T^{2} - 24671264664594287440 T^{3} + 48614168717389 p^{7} T^{4} - 1004504 p^{14} T^{5} + p^{21} T^{6} )^{2} \) |
| 83 | \( 1 - 77901835470102 T^{2} + \)\(36\!\cdots\!95\)\( T^{4} - \)\(11\!\cdots\!16\)\( T^{6} + \)\(36\!\cdots\!95\)\( p^{14} T^{8} - 77901835470102 p^{28} T^{10} + p^{42} T^{12} \) |
| 89 | \( ( 1 - 4831350 T + 110343597017031 T^{2} - \)\(33\!\cdots\!76\)\( T^{3} + 110343597017031 p^{7} T^{4} - 4831350 p^{14} T^{5} + p^{21} T^{6} )^{2} \) |
| 97 | \( ( 1 + 5250694 T + 46471338379759 T^{2} + 10300446712993234580 T^{3} + 46471338379759 p^{7} T^{4} + 5250694 p^{14} T^{5} + p^{21} T^{6} )^{2} \) |
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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
−5.12892760076925434994803229187, −4.93276775865314288038220559148, −4.91193811228021640405599213749, −4.73901450322078396936279969653, −4.42879412310107096566565248408, −4.31831767471558219213480091187, −4.04674004351708643702997407621, −3.94424340210558837047532708383, −3.90683155711702791231074436479, −3.72137322080117082721512014660, −3.08470549000108652797562783034, −2.95580872982943577929183769152, −2.91048497680093999802850724531, −2.69699166460437785319024003050, −2.41852129632272313688756208279, −2.38120768253847334088560743344, −1.75359359982016601089161520149, −1.62946115226192009252318427772, −1.57124592491336893850328830227, −1.19490742575214262605499157746, −1.09891654564282015336339503752, −1.06378763945646621436039725439, −0.66532081652600839736248434986, −0.35432588433021781826754136363, −0.30116304250720676260863352603,
0.30116304250720676260863352603, 0.35432588433021781826754136363, 0.66532081652600839736248434986, 1.06378763945646621436039725439, 1.09891654564282015336339503752, 1.19490742575214262605499157746, 1.57124592491336893850328830227, 1.62946115226192009252318427772, 1.75359359982016601089161520149, 2.38120768253847334088560743344, 2.41852129632272313688756208279, 2.69699166460437785319024003050, 2.91048497680093999802850724531, 2.95580872982943577929183769152, 3.08470549000108652797562783034, 3.72137322080117082721512014660, 3.90683155711702791231074436479, 3.94424340210558837047532708383, 4.04674004351708643702997407621, 4.31831767471558219213480091187, 4.42879412310107096566565248408, 4.73901450322078396936279969653, 4.91193811228021640405599213749, 4.93276775865314288038220559148, 5.12892760076925434994803229187
Plot not available for L-functions of degree greater than 10.
