| L(s) = 1 | − 56·3-s + 319·4-s − 5.56e3·9-s − 1.78e4·12-s − 5.01e3·13-s + 4.86e4·16-s + 1.31e4·17-s + 2.72e4·23-s + 2.25e5·25-s + 4.76e5·27-s + 4.29e4·29-s − 1.77e6·36-s + 2.81e5·39-s − 1.00e6·43-s − 2.72e6·48-s + 4.09e6·49-s − 7.36e5·51-s − 1.60e6·52-s + 1.70e6·53-s − 8.23e6·61-s + 5.71e6·64-s + 4.19e6·68-s − 1.52e6·69-s − 1.26e7·75-s + 1.51e7·79-s + 4.06e6·81-s − 2.40e6·87-s + ⋯ |
| L(s) = 1 | − 1.19·3-s + 2.49·4-s − 2.54·9-s − 2.98·12-s − 0.633·13-s + 2.97·16-s + 0.649·17-s + 0.467·23-s + 2.88·25-s + 4.65·27-s + 0.326·29-s − 6.34·36-s + 0.758·39-s − 1.92·43-s − 3.55·48-s + 4.97·49-s − 0.777·51-s − 1.57·52-s + 1.57·53-s − 4.64·61-s + 2.72·64-s + 1.61·68-s − 0.559·69-s − 3.45·75-s + 3.45·79-s + 0.849·81-s − 0.391·87-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4826809 ^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4826809 ^{s/2} \, \Gamma_{\C}(s+7/2)^{6} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(3.198690546\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.198690546\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 13 | \( 1 + 386 p T + 44051 p^{3} T^{2} + 168260 p^{6} T^{3} + 44051 p^{10} T^{4} + 386 p^{15} T^{5} + p^{21} T^{6} \) |
| good | 2 | \( 1 - 319 T^{2} + 6637 p^{3} T^{4} - 445507 p^{4} T^{6} + 6637 p^{17} T^{8} - 319 p^{28} T^{10} + p^{42} T^{12} \) |
| 3 | \( ( 1 + 28 T + 440 p^{2} T^{2} + 5636 p^{2} T^{3} + 440 p^{9} T^{4} + 28 p^{14} T^{5} + p^{21} T^{6} )^{2} \) |
| 5 | \( 1 - 225244 T^{2} + 26661632 p^{4} T^{4} - 1292312020546 p^{4} T^{6} + 26661632 p^{18} T^{8} - 225244 p^{28} T^{10} + p^{42} T^{12} \) |
| 7 | \( 1 - 4094052 T^{2} + 7613928151728 T^{4} - 8084053501544371826 T^{6} + 7613928151728 p^{14} T^{8} - 4094052 p^{28} T^{10} + p^{42} T^{12} \) |
| 11 | \( 1 - 40509598 T^{2} + 1553203332107159 T^{4} - \)\(30\!\cdots\!64\)\( T^{6} + 1553203332107159 p^{14} T^{8} - 40509598 p^{28} T^{10} + p^{42} T^{12} \) |
| 17 | \( ( 1 - 6576 T + 670890984 T^{2} - 6373405665546 T^{3} + 670890984 p^{7} T^{4} - 6576 p^{14} T^{5} + p^{21} T^{6} )^{2} \) |
| 19 | \( 1 - 4171429446 T^{2} + 8197248268449992535 T^{4} - \)\(93\!\cdots\!00\)\( T^{6} + 8197248268449992535 p^{14} T^{8} - 4171429446 p^{28} T^{10} + p^{42} T^{12} \) |
| 23 | \( ( 1 - 13632 T + 6634217697 T^{2} - 131389967983200 T^{3} + 6634217697 p^{7} T^{4} - 13632 p^{14} T^{5} + p^{21} T^{6} )^{2} \) |
| 29 | \( ( 1 - 21462 T + 23484408783 T^{2} + 938622857665764 T^{3} + 23484408783 p^{7} T^{4} - 21462 p^{14} T^{5} + p^{21} T^{6} )^{2} \) |
| 31 | \( 1 - 87189975654 T^{2} + \)\(38\!\cdots\!23\)\( T^{4} - \)\(11\!\cdots\!00\)\( T^{6} + \)\(38\!\cdots\!23\)\( p^{14} T^{8} - 87189975654 p^{28} T^{10} + p^{42} T^{12} \) |
| 37 | \( 1 + 40604175876 T^{2} + \)\(64\!\cdots\!96\)\( T^{4} + \)\(48\!\cdots\!58\)\( T^{6} + \)\(64\!\cdots\!96\)\( p^{14} T^{8} + 40604175876 p^{28} T^{10} + p^{42} T^{12} \) |
| 41 | \( 1 - 981327751318 T^{2} + \)\(42\!\cdots\!39\)\( T^{4} - \)\(10\!\cdots\!04\)\( T^{6} + \)\(42\!\cdots\!39\)\( p^{14} T^{8} - 981327751318 p^{28} T^{10} + p^{42} T^{12} \) |
| 43 | \( ( 1 + 502788 T + 741662050272 T^{2} + 274465314975443020 T^{3} + 741662050272 p^{7} T^{4} + 502788 p^{14} T^{5} + p^{21} T^{6} )^{2} \) |
| 47 | \( 1 - 1332238936180 T^{2} + \)\(85\!\cdots\!32\)\( T^{4} - \)\(40\!\cdots\!90\)\( T^{6} + \)\(85\!\cdots\!32\)\( p^{14} T^{8} - 1332238936180 p^{28} T^{10} + p^{42} T^{12} \) |
| 53 | \( ( 1 - 852762 T + 2759104557555 T^{2} - 1820850683685395196 T^{3} + 2759104557555 p^{7} T^{4} - 852762 p^{14} T^{5} + p^{21} T^{6} )^{2} \) |
| 59 | \( 1 - 8151276545110 T^{2} + \)\(28\!\cdots\!35\)\( T^{4} - \)\(72\!\cdots\!32\)\( T^{6} + \)\(28\!\cdots\!35\)\( p^{14} T^{8} - 8151276545110 p^{28} T^{10} + p^{42} T^{12} \) |
| 61 | \( ( 1 + 4118786 T + 12195040984463 T^{2} + 25992513319044665300 T^{3} + 12195040984463 p^{7} T^{4} + 4118786 p^{14} T^{5} + p^{21} T^{6} )^{2} \) |
| 67 | \( 1 - 9560927123790 T^{2} + \)\(12\!\cdots\!87\)\( T^{4} - \)\(69\!\cdots\!20\)\( T^{6} + \)\(12\!\cdots\!87\)\( p^{14} T^{8} - 9560927123790 p^{28} T^{10} + p^{42} T^{12} \) |
| 71 | \( 1 - 26778276858868 T^{2} + \)\(42\!\cdots\!44\)\( T^{4} - \)\(44\!\cdots\!34\)\( T^{6} + \)\(42\!\cdots\!44\)\( p^{14} T^{8} - 26778276858868 p^{28} T^{10} + p^{42} T^{12} \) |
| 73 | \( 1 - 31732514734830 T^{2} + \)\(66\!\cdots\!27\)\( T^{4} - \)\(84\!\cdots\!40\)\( T^{6} + \)\(66\!\cdots\!27\)\( p^{14} T^{8} - 31732514734830 p^{28} T^{10} + p^{42} T^{12} \) |
| 79 | \( ( 1 - 7573600 T + 75273279006377 T^{2} - \)\(30\!\cdots\!00\)\( T^{3} + 75273279006377 p^{7} T^{4} - 7573600 p^{14} T^{5} + p^{21} T^{6} )^{2} \) |
| 83 | \( 1 - 80496451585030 T^{2} + \)\(33\!\cdots\!87\)\( T^{4} - \)\(10\!\cdots\!40\)\( T^{6} + \)\(33\!\cdots\!87\)\( p^{14} T^{8} - 80496451585030 p^{28} T^{10} + p^{42} T^{12} \) |
| 89 | \( 1 - 78847341921070 T^{2} + \)\(77\!\cdots\!75\)\( T^{4} - \)\(32\!\cdots\!92\)\( T^{6} + \)\(77\!\cdots\!75\)\( p^{14} T^{8} - 78847341921070 p^{28} T^{10} + p^{42} T^{12} \) |
| 97 | \( 1 - 132335230608294 T^{2} + \)\(67\!\cdots\!51\)\( T^{4} - \)\(41\!\cdots\!92\)\( T^{6} + \)\(67\!\cdots\!51\)\( p^{14} T^{8} - 132335230608294 p^{28} T^{10} + p^{42} T^{12} \) |
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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
−10.41728212494074611936827364797, −10.39693235349534933812910087056, −9.911969504320563643371150670113, −9.129547907601130901687093152870, −8.907596455195162762939754818786, −8.735681886020647477304056589957, −8.679170094350842031203687605705, −7.84038819283196400533718859534, −7.81945150775814257357244166822, −7.17942133829713314136233129354, −7.13524546219852640362098146785, −6.53348588237262236850057329325, −6.52548784695551648180895337821, −5.98959406007401701645779062436, −5.85903372389327096984658518377, −5.42037893774710437139966590847, −4.93666054130153860427867968019, −4.90659799220475828249320606046, −3.73335292189720968711042580781, −2.99806685421999627384279384757, −2.80224878401327358655124585665, −2.65992358500043572121106935725, −1.92684499341758992252271506019, −0.892437544707419309294015539528, −0.54865171335517690371365106374,
0.54865171335517690371365106374, 0.892437544707419309294015539528, 1.92684499341758992252271506019, 2.65992358500043572121106935725, 2.80224878401327358655124585665, 2.99806685421999627384279384757, 3.73335292189720968711042580781, 4.90659799220475828249320606046, 4.93666054130153860427867968019, 5.42037893774710437139966590847, 5.85903372389327096984658518377, 5.98959406007401701645779062436, 6.52548784695551648180895337821, 6.53348588237262236850057329325, 7.13524546219852640362098146785, 7.17942133829713314136233129354, 7.81945150775814257357244166822, 7.84038819283196400533718859534, 8.679170094350842031203687605705, 8.735681886020647477304056589957, 8.907596455195162762939754818786, 9.129547907601130901687093152870, 9.911969504320563643371150670113, 10.39693235349534933812910087056, 10.41728212494074611936827364797
Plot not available for L-functions of degree greater than 10.
