| L(s) = 1 | − 2-s + 9·3-s − 8·4-s − 9·6-s − 16·7-s + 12·8-s + 54·9-s + 33·11-s − 72·12-s − 45·13-s + 16·14-s − 7·16-s + 58·17-s − 54·18-s − 169·19-s − 144·21-s − 33·22-s − 155·23-s + 108·24-s + 45·26-s + 270·27-s + 128·28-s − 277·29-s − 173·31-s − 53·32-s + 297·33-s − 58·34-s + ⋯ |
| L(s) = 1 | − 0.353·2-s + 1.73·3-s − 4-s − 0.612·6-s − 0.863·7-s + 0.530·8-s + 2·9-s + 0.904·11-s − 1.73·12-s − 0.960·13-s + 0.305·14-s − 0.109·16-s + 0.827·17-s − 0.707·18-s − 2.04·19-s − 1.49·21-s − 0.319·22-s − 1.40·23-s + 0.918·24-s + 0.339·26-s + 1.92·27-s + 0.863·28-s − 1.77·29-s − 1.00·31-s − 0.292·32-s + 1.56·33-s − 0.292·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{3} \cdot 5^{6} \cdot 11^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{3} \cdot 5^{6} \cdot 11^{3}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
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 | $C_1$ | \( ( 1 - p T )^{3} \) |
| 5 | | \( 1 \) |
| 11 | $C_1$ | \( ( 1 - p T )^{3} \) |
| good | 2 | $S_4\times C_2$ | \( 1 + T + 9 T^{2} + 5 T^{3} + 9 p^{3} T^{4} + p^{6} T^{5} + p^{9} T^{6} \) |
| 7 | $S_4\times C_2$ | \( 1 + 16 T + 856 T^{2} + 11150 T^{3} + 856 p^{3} T^{4} + 16 p^{6} T^{5} + p^{9} T^{6} \) |
| 13 | $S_4\times C_2$ | \( 1 + 45 T + 6871 T^{2} + 192794 T^{3} + 6871 p^{3} T^{4} + 45 p^{6} T^{5} + p^{9} T^{6} \) |
| 17 | $S_4\times C_2$ | \( 1 - 58 T + 15434 T^{2} - 572224 T^{3} + 15434 p^{3} T^{4} - 58 p^{6} T^{5} + p^{9} T^{6} \) |
| 19 | $S_4\times C_2$ | \( 1 + 169 T + 28098 T^{2} + 2360147 T^{3} + 28098 p^{3} T^{4} + 169 p^{6} T^{5} + p^{9} T^{6} \) |
| 23 | $S_4\times C_2$ | \( 1 + 155 T + 41944 T^{2} + 3812131 T^{3} + 41944 p^{3} T^{4} + 155 p^{6} T^{5} + p^{9} T^{6} \) |
| 29 | $S_4\times C_2$ | \( 1 + 277 T + 68543 T^{2} + 12078682 T^{3} + 68543 p^{3} T^{4} + 277 p^{6} T^{5} + p^{9} T^{6} \) |
| 31 | $S_4\times C_2$ | \( 1 + 173 T + 42261 T^{2} + 6587230 T^{3} + 42261 p^{3} T^{4} + 173 p^{6} T^{5} + p^{9} T^{6} \) |
| 37 | $S_4\times C_2$ | \( 1 + 60 T + 3724 T^{2} - 9069062 T^{3} + 3724 p^{3} T^{4} + 60 p^{6} T^{5} + p^{9} T^{6} \) |
| 41 | $S_4\times C_2$ | \( 1 - 44 T + 133902 T^{2} - 3107666 T^{3} + 133902 p^{3} T^{4} - 44 p^{6} T^{5} + p^{9} T^{6} \) |
| 43 | $S_4\times C_2$ | \( 1 - 109 T + 228873 T^{2} - 16964786 T^{3} + 228873 p^{3} T^{4} - 109 p^{6} T^{5} + p^{9} T^{6} \) |
| 47 | $S_4\times C_2$ | \( 1 + 270 T + 208054 T^{2} + 52560216 T^{3} + 208054 p^{3} T^{4} + 270 p^{6} T^{5} + p^{9} T^{6} \) |
| 53 | $S_4\times C_2$ | \( 1 - 148 T + 385579 T^{2} - 33949544 T^{3} + 385579 p^{3} T^{4} - 148 p^{6} T^{5} + p^{9} T^{6} \) |
| 59 | $S_4\times C_2$ | \( 1 + 684 T + 535726 T^{2} + 196887298 T^{3} + 535726 p^{3} T^{4} + 684 p^{6} T^{5} + p^{9} T^{6} \) |
| 61 | $S_4\times C_2$ | \( 1 + 1038 T + 631795 T^{2} + 333327140 T^{3} + 631795 p^{3} T^{4} + 1038 p^{6} T^{5} + p^{9} T^{6} \) |
| 67 | $S_4\times C_2$ | \( 1 + 314 T + 854861 T^{2} + 187279700 T^{3} + 854861 p^{3} T^{4} + 314 p^{6} T^{5} + p^{9} T^{6} \) |
| 71 | $S_4\times C_2$ | \( 1 + 1459 T + 1463380 T^{2} + 997438583 T^{3} + 1463380 p^{3} T^{4} + 1459 p^{6} T^{5} + p^{9} T^{6} \) |
| 73 | $S_4\times C_2$ | \( 1 - 1170 T + 1393615 T^{2} - 839920876 T^{3} + 1393615 p^{3} T^{4} - 1170 p^{6} T^{5} + p^{9} T^{6} \) |
| 79 | $S_4\times C_2$ | \( 1 + 506 T + 1295660 T^{2} + 507308816 T^{3} + 1295660 p^{3} T^{4} + 506 p^{6} T^{5} + p^{9} T^{6} \) |
| 83 | $S_4\times C_2$ | \( 1 - 347 T + 1599297 T^{2} - 390027914 T^{3} + 1599297 p^{3} T^{4} - 347 p^{6} T^{5} + p^{9} T^{6} \) |
| 89 | $S_4\times C_2$ | \( 1 + 607 T + 1367067 T^{2} + 593095898 T^{3} + 1367067 p^{3} T^{4} + 607 p^{6} T^{5} + p^{9} T^{6} \) |
| 97 | $S_4\times C_2$ | \( 1 + 1263 T + 3146554 T^{2} + 2315882599 T^{3} + 3146554 p^{3} T^{4} + 1263 p^{6} T^{5} + p^{9} T^{6} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{6} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.303096338694315331886966347972, −8.755682507851789736443528132673, −8.616664110169027445670224872884, −8.576179336098556867510594689081, −7.962484579762390112228576164610, −7.82620663025620707697244771930, −7.42569176591906898709597473041, −7.36543990800846358327776663057, −6.99754085167477805901229589265, −6.52735723869503014324376411966, −6.20075512470464969751454236091, −5.89409773024675972630581253078, −5.85866678728864268153075339313, −4.87705170518997450586048041319, −4.76099205744128501497339822617, −4.66974320855044842980915030680, −4.03340274172898315513678322146, −3.72804636049900806135500907917, −3.70850380636525562155873767261, −3.25053997493087988360986984978, −2.78583340557904427382802754752, −2.31405420026820104086321520550, −2.03903227009957904608742690726, −1.43239054369412809949472506116, −1.40967291240188729297178882749, 0, 0, 0,
1.40967291240188729297178882749, 1.43239054369412809949472506116, 2.03903227009957904608742690726, 2.31405420026820104086321520550, 2.78583340557904427382802754752, 3.25053997493087988360986984978, 3.70850380636525562155873767261, 3.72804636049900806135500907917, 4.03340274172898315513678322146, 4.66974320855044842980915030680, 4.76099205744128501497339822617, 4.87705170518997450586048041319, 5.85866678728864268153075339313, 5.89409773024675972630581253078, 6.20075512470464969751454236091, 6.52735723869503014324376411966, 6.99754085167477805901229589265, 7.36543990800846358327776663057, 7.42569176591906898709597473041, 7.82620663025620707697244771930, 7.962484579762390112228576164610, 8.576179336098556867510594689081, 8.616664110169027445670224872884, 8.755682507851789736443528132673, 9.303096338694315331886966347972
