| L(s) = 1 | + 2·2-s + 9·3-s − 5·4-s + 4·5-s + 18·6-s + 30·7-s − 20·8-s + 54·9-s + 8·10-s − 16·11-s − 45·12-s + 39·13-s + 60·14-s + 36·15-s − 51·16-s − 146·17-s + 108·18-s + 94·19-s − 20·20-s + 270·21-s − 32·22-s − 48·23-s − 180·24-s − 107·25-s + 78·26-s + 270·27-s − 150·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1.73·3-s − 5/8·4-s + 0.357·5-s + 1.22·6-s + 1.61·7-s − 0.883·8-s + 2·9-s + 0.252·10-s − 0.438·11-s − 1.08·12-s + 0.832·13-s + 1.14·14-s + 0.619·15-s − 0.796·16-s − 2.08·17-s + 1.41·18-s + 1.13·19-s − 0.223·20-s + 2.80·21-s − 0.310·22-s − 0.435·23-s − 1.53·24-s − 0.855·25-s + 0.588·26-s + 1.92·27-s − 1.01·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 59319 ^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 59319 ^{s/2} \, \Gamma_{\C}(s+3/2)^{3} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(4.344688956\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.344688956\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | $C_1$ | \( ( 1 - p T )^{3} \) |
| 13 | $C_1$ | \( ( 1 - p T )^{3} \) |
| good | 2 | $S_4\times C_2$ | \( 1 - p T + 9 T^{2} - p^{3} T^{3} + 9 p^{3} T^{4} - p^{7} T^{5} + p^{9} T^{6} \) |
| 5 | $S_4\times C_2$ | \( 1 - 4 T + 123 T^{2} - 1864 T^{3} + 123 p^{3} T^{4} - 4 p^{6} T^{5} + p^{9} T^{6} \) |
| 7 | $S_4\times C_2$ | \( 1 - 30 T + 741 T^{2} - 18596 T^{3} + 741 p^{3} T^{4} - 30 p^{6} T^{5} + p^{9} T^{6} \) |
| 11 | $S_4\times C_2$ | \( 1 + 16 T + 1737 T^{2} + 72928 T^{3} + 1737 p^{3} T^{4} + 16 p^{6} T^{5} + p^{9} T^{6} \) |
| 17 | $S_4\times C_2$ | \( 1 + 146 T + 20799 T^{2} + 1505852 T^{3} + 20799 p^{3} T^{4} + 146 p^{6} T^{5} + p^{9} T^{6} \) |
| 19 | $S_4\times C_2$ | \( 1 - 94 T + 6145 T^{2} - 509876 T^{3} + 6145 p^{3} T^{4} - 94 p^{6} T^{5} + p^{9} T^{6} \) |
| 23 | $S_4\times C_2$ | \( 1 + 48 T + 15573 T^{2} + 1702560 T^{3} + 15573 p^{3} T^{4} + 48 p^{6} T^{5} + p^{9} T^{6} \) |
| 29 | $S_4\times C_2$ | \( 1 + 2 T + 63051 T^{2} - 101620 T^{3} + 63051 p^{3} T^{4} + 2 p^{6} T^{5} + p^{9} T^{6} \) |
| 31 | $S_4\times C_2$ | \( 1 - 302 T + 71837 T^{2} - 10796516 T^{3} + 71837 p^{3} T^{4} - 302 p^{6} T^{5} + p^{9} T^{6} \) |
| 37 | $S_4\times C_2$ | \( 1 - 374 T + 114995 T^{2} - 30130340 T^{3} + 114995 p^{3} T^{4} - 374 p^{6} T^{5} + p^{9} T^{6} \) |
| 41 | $S_4\times C_2$ | \( 1 - 480 T + 208479 T^{2} - 53244336 T^{3} + 208479 p^{3} T^{4} - 480 p^{6} T^{5} + p^{9} T^{6} \) |
| 43 | $S_4\times C_2$ | \( 1 + 260 T + 200425 T^{2} + 37680472 T^{3} + 200425 p^{3} T^{4} + 260 p^{6} T^{5} + p^{9} T^{6} \) |
| 47 | $S_4\times C_2$ | \( 1 + 24 T + 142989 T^{2} + 23086032 T^{3} + 142989 p^{3} T^{4} + 24 p^{6} T^{5} + p^{9} T^{6} \) |
| 53 | $S_4\times C_2$ | \( 1 + 678 T + 404403 T^{2} + 200405604 T^{3} + 404403 p^{3} T^{4} + 678 p^{6} T^{5} + p^{9} T^{6} \) |
| 59 | $S_4\times C_2$ | \( 1 + 1788 T + 1572249 T^{2} + 871859112 T^{3} + 1572249 p^{3} T^{4} + 1788 p^{6} T^{5} + p^{9} T^{6} \) |
| 61 | $S_4\times C_2$ | \( 1 - 230 T + 636491 T^{2} - 98131748 T^{3} + 636491 p^{3} T^{4} - 230 p^{6} T^{5} + p^{9} T^{6} \) |
| 67 | $S_4\times C_2$ | \( 1 - 74 T + 493073 T^{2} - 40252028 T^{3} + 493073 p^{3} T^{4} - 74 p^{6} T^{5} + p^{9} T^{6} \) |
| 71 | $S_4\times C_2$ | \( 1 + 948 T + 1061157 T^{2} + 608134872 T^{3} + 1061157 p^{3} T^{4} + 948 p^{6} T^{5} + p^{9} T^{6} \) |
| 73 | $S_4\times C_2$ | \( 1 + 222 T + 223815 T^{2} + 195504100 T^{3} + 223815 p^{3} T^{4} + 222 p^{6} T^{5} + p^{9} T^{6} \) |
| 79 | $S_4\times C_2$ | \( 1 + 24 T + 1400781 T^{2} + 31423696 T^{3} + 1400781 p^{3} T^{4} + 24 p^{6} T^{5} + p^{9} T^{6} \) |
| 83 | $S_4\times C_2$ | \( 1 + 796 T + 1904433 T^{2} + 924248872 T^{3} + 1904433 p^{3} T^{4} + 796 p^{6} T^{5} + p^{9} T^{6} \) |
| 89 | $S_4\times C_2$ | \( 1 - 1436 T + 2536191 T^{2} - 2054800856 T^{3} + 2536191 p^{3} T^{4} - 1436 p^{6} T^{5} + p^{9} T^{6} \) |
| 97 | $S_4\times C_2$ | \( 1 - 3242 T + 6203519 T^{2} - 7136252780 T^{3} + 6203519 p^{3} T^{4} - 3242 p^{6} T^{5} + p^{9} T^{6} \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{6} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−14.30044286038327071267886798589, −13.63650315395773307900783175105, −13.57424298072454794785598028302, −13.25708409642756462427154423655, −13.00245693476129854720288499989, −12.28893091620450122361463384680, −11.60312994981776298364481036927, −11.52880811330999102287071951704, −10.76958401161466343610986832455, −10.57736556386613563650598886951, −9.502694959866380525191190996779, −9.502178898180561564865884871245, −9.125313057728331244327295481628, −8.329304548051192620647923242570, −8.305794915497233695757295811161, −7.76775780652918002097914124192, −7.32085853754735666088643676370, −6.23769169350368111799495865864, −6.09547993294305422456410394519, −4.81817224080894136526122061989, −4.56649153778551490017582005643, −4.30788066429207028148735388895, −3.22991637083297119472160624174, −2.50142958496538768405667615354, −1.61182073826344544636161528361,
1.61182073826344544636161528361, 2.50142958496538768405667615354, 3.22991637083297119472160624174, 4.30788066429207028148735388895, 4.56649153778551490017582005643, 4.81817224080894136526122061989, 6.09547993294305422456410394519, 6.23769169350368111799495865864, 7.32085853754735666088643676370, 7.76775780652918002097914124192, 8.305794915497233695757295811161, 8.329304548051192620647923242570, 9.125313057728331244327295481628, 9.502178898180561564865884871245, 9.502694959866380525191190996779, 10.57736556386613563650598886951, 10.76958401161466343610986832455, 11.52880811330999102287071951704, 11.60312994981776298364481036927, 12.28893091620450122361463384680, 13.00245693476129854720288499989, 13.25708409642756462427154423655, 13.57424298072454794785598028302, 13.63650315395773307900783175105, 14.30044286038327071267886798589
