| L(s) = 1 | − 2-s − 8·4-s + 16·7-s + 12·8-s − 33·11-s + 45·13-s − 16·14-s − 7·16-s + 58·17-s − 169·19-s + 33·22-s − 155·23-s − 45·26-s − 128·28-s + 277·29-s − 173·31-s − 53·32-s − 58·34-s + 60·37-s + 169·38-s − 44·41-s − 109·43-s + 264·44-s + 155·46-s − 270·47-s − 600·49-s − 360·52-s + ⋯ |
| L(s) = 1 | − 0.353·2-s − 4-s + 0.863·7-s + 0.530·8-s − 0.904·11-s + 0.960·13-s − 0.305·14-s − 0.109·16-s + 0.827·17-s − 2.04·19-s + 0.319·22-s − 1.40·23-s − 0.339·26-s − 0.863·28-s + 1.77·29-s − 1.00·31-s − 0.292·32-s − 0.292·34-s + 0.266·37-s + 0.721·38-s − 0.167·41-s − 0.386·43-s + 0.904·44-s + 0.496·46-s − 0.837·47-s − 1.74·49-s − 0.960·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{6} \cdot 5^{6} \cdot 11^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{6} \cdot 5^{6} \cdot 11^{3}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]
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 | | \( 1 \) |
| 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
−8.118317832921700573298589410104, −7.70404876003238400828711026515, −7.68689843590326600292642929956, −7.33304568780582597426060088346, −6.85983408587648447115154863048, −6.60156393568827618328371755523, −6.40070387776346695537497493813, −6.16322256798716487300231275938, −5.87006247369572543624364456082, −5.61497713994008519672382193005, −5.14508688907335366592253377243, −4.93216022047827053506083889211, −4.91837487161020108513765943130, −4.37735124843775149016248548822, −4.23634240468506285450847731495, −4.08011478932058509051533724222, −3.48787369225175766709524233710, −3.40810718077426570558115688269, −3.04222779260184957883982800014, −2.46813537405271466369895016293, −2.20645791519083836536642427676, −1.94673506044199708653465687081, −1.58657254811156824820093674793, −1.01470169380167116607799455431, −0.997909905006072522647684163779, 0, 0, 0,
0.997909905006072522647684163779, 1.01470169380167116607799455431, 1.58657254811156824820093674793, 1.94673506044199708653465687081, 2.20645791519083836536642427676, 2.46813537405271466369895016293, 3.04222779260184957883982800014, 3.40810718077426570558115688269, 3.48787369225175766709524233710, 4.08011478932058509051533724222, 4.23634240468506285450847731495, 4.37735124843775149016248548822, 4.91837487161020108513765943130, 4.93216022047827053506083889211, 5.14508688907335366592253377243, 5.61497713994008519672382193005, 5.87006247369572543624364456082, 6.16322256798716487300231275938, 6.40070387776346695537497493813, 6.60156393568827618328371755523, 6.85983408587648447115154863048, 7.33304568780582597426060088346, 7.68689843590326600292642929956, 7.70404876003238400828711026515, 8.118317832921700573298589410104
