| L(s) = 1 | + 2·2-s + 9·3-s + 5·4-s − 15·5-s + 18·6-s − 10·7-s − 4·8-s + 54·9-s − 30·10-s + 45·12-s − 114·13-s − 20·14-s − 135·15-s − 27·16-s + 104·17-s + 108·18-s + 58·19-s − 75·20-s − 90·21-s + 120·23-s − 36·24-s + 150·25-s − 228·26-s + 270·27-s − 50·28-s + 220·29-s − 270·30-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1.73·3-s + 5/8·4-s − 1.34·5-s + 1.22·6-s − 0.539·7-s − 0.176·8-s + 2·9-s − 0.948·10-s + 1.08·12-s − 2.43·13-s − 0.381·14-s − 2.32·15-s − 0.421·16-s + 1.48·17-s + 1.41·18-s + 0.700·19-s − 0.838·20-s − 0.935·21-s + 1.08·23-s − 0.306·24-s + 6/5·25-s − 1.71·26-s + 1.92·27-s − 0.337·28-s + 1.40·29-s − 1.64·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{3} \cdot 5^{3} \cdot 11^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{3} \cdot 5^{3} \cdot 11^{6}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(16.75731072\) |
| \(L(\frac12)\) |
\(\approx\) |
\(16.75731072\) |
| \(L(\frac{5}{2})\) |
|
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 | $C_1$ | \( ( 1 + p T )^{3} \) |
| 11 | | \( 1 \) |
| good | 2 | $S_4\times C_2$ | \( 1 - p T - T^{2} + p^{4} T^{3} - p^{3} T^{4} - p^{7} T^{5} + p^{9} T^{6} \) |
| 7 | $S_4\times C_2$ | \( 1 + 10 T + 425 T^{2} + 3412 T^{3} + 425 p^{3} T^{4} + 10 p^{6} T^{5} + p^{9} T^{6} \) |
| 13 | $S_4\times C_2$ | \( 1 + 114 T + 10303 T^{2} + 538132 T^{3} + 10303 p^{3} T^{4} + 114 p^{6} T^{5} + p^{9} T^{6} \) |
| 17 | $S_4\times C_2$ | \( 1 - 104 T + 17531 T^{2} - 1030352 T^{3} + 17531 p^{3} T^{4} - 104 p^{6} T^{5} + p^{9} T^{6} \) |
| 19 | $S_4\times C_2$ | \( 1 - 58 T + 9081 T^{2} - 861164 T^{3} + 9081 p^{3} T^{4} - 58 p^{6} T^{5} + p^{9} T^{6} \) |
| 23 | $S_4\times C_2$ | \( 1 - 120 T + 25765 T^{2} - 2771856 T^{3} + 25765 p^{3} T^{4} - 120 p^{6} T^{5} + p^{9} T^{6} \) |
| 29 | $S_4\times C_2$ | \( 1 - 220 T + 58859 T^{2} - 10101400 T^{3} + 58859 p^{3} T^{4} - 220 p^{6} T^{5} + p^{9} T^{6} \) |
| 31 | $S_4\times C_2$ | \( 1 - 8 p T + 50781 T^{2} - 5187088 T^{3} + 50781 p^{3} T^{4} - 8 p^{7} T^{5} + p^{9} T^{6} \) |
| 37 | $S_4\times C_2$ | \( 1 - 838 T + 375507 T^{2} - 103501764 T^{3} + 375507 p^{3} T^{4} - 838 p^{6} T^{5} + p^{9} T^{6} \) |
| 41 | $S_4\times C_2$ | \( 1 + 156 T + 100167 T^{2} + 24516984 T^{3} + 100167 p^{3} T^{4} + 156 p^{6} T^{5} + p^{9} T^{6} \) |
| 43 | $S_4\times C_2$ | \( 1 + 122 T + 193581 T^{2} + 17954308 T^{3} + 193581 p^{3} T^{4} + 122 p^{6} T^{5} + p^{9} T^{6} \) |
| 47 | $S_4\times C_2$ | \( 1 - 504 T + 393661 T^{2} - 109025808 T^{3} + 393661 p^{3} T^{4} - 504 p^{6} T^{5} + p^{9} T^{6} \) |
| 53 | $S_4\times C_2$ | \( 1 - 282 T + 224563 T^{2} - 87620892 T^{3} + 224563 p^{3} T^{4} - 282 p^{6} T^{5} + p^{9} T^{6} \) |
| 59 | $S_4\times C_2$ | \( 1 - 548 T + 11419 p T^{2} - 226302104 T^{3} + 11419 p^{4} T^{4} - 548 p^{6} T^{5} + p^{9} T^{6} \) |
| 61 | $S_4\times C_2$ | \( 1 + 414 T - 389 T^{2} - 154404524 T^{3} - 389 p^{3} T^{4} + 414 p^{6} T^{5} + p^{9} T^{6} \) |
| 67 | $S_4\times C_2$ | \( 1 + 428 T + 695537 T^{2} + 249317576 T^{3} + 695537 p^{3} T^{4} + 428 p^{6} T^{5} + p^{9} T^{6} \) |
| 71 | $S_4\times C_2$ | \( 1 + 912 T + 1171621 T^{2} + 649961952 T^{3} + 1171621 p^{3} T^{4} + 912 p^{6} T^{5} + p^{9} T^{6} \) |
| 73 | $S_4\times C_2$ | \( 1 + 618 T + 1066507 T^{2} + 454366420 T^{3} + 1066507 p^{3} T^{4} + 618 p^{6} T^{5} + p^{9} T^{6} \) |
| 79 | $S_4\times C_2$ | \( 1 - 542 T + 1317893 T^{2} - 445950836 T^{3} + 1317893 p^{3} T^{4} - 542 p^{6} T^{5} + p^{9} T^{6} \) |
| 83 | $S_4\times C_2$ | \( 1 + 624021 T^{2} + 434328048 T^{3} + 624021 p^{3} T^{4} + p^{9} T^{6} \) |
| 89 | $S_4\times C_2$ | \( 1 - 790 T - 3625 T^{2} + 827778380 T^{3} - 3625 p^{3} T^{4} - 790 p^{6} T^{5} + p^{9} T^{6} \) |
| 97 | $S_4\times C_2$ | \( 1 - 2074 T + 3705231 T^{2} - 3687692268 T^{3} + 3705231 p^{3} T^{4} - 2074 p^{6} T^{5} + p^{9} T^{6} \) |
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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
−7.68155212738953520809810581499, −7.56538731061693665212905206945, −7.50155607756184202835593213827, −7.26307822358259254531316984862, −6.93837220696184295005558493422, −6.48706647654754191234602591050, −6.33738140739165496773939797422, −6.13353153738808215739894937211, −5.58428701515979438719512434920, −5.33969528692438469881444665079, −4.78658643954611606608651611457, −4.68295404203727403511285468873, −4.58896284724051986410151801347, −4.22751270704659360203077457517, −3.81745474533621915538725886623, −3.50939886386724847681186909810, −3.00685929005974086106707187204, −2.99298786691303386945408748121, −2.75192513907620275509136818853, −2.64072958350920720113026199117, −2.15098337963056149801790933048, −1.61778701123934422766480284896, −0.977004267892786146990232190982, −0.68932957538676301339132156075, −0.53196175745247068709289621720,
0.53196175745247068709289621720, 0.68932957538676301339132156075, 0.977004267892786146990232190982, 1.61778701123934422766480284896, 2.15098337963056149801790933048, 2.64072958350920720113026199117, 2.75192513907620275509136818853, 2.99298786691303386945408748121, 3.00685929005974086106707187204, 3.50939886386724847681186909810, 3.81745474533621915538725886623, 4.22751270704659360203077457517, 4.58896284724051986410151801347, 4.68295404203727403511285468873, 4.78658643954611606608651611457, 5.33969528692438469881444665079, 5.58428701515979438719512434920, 6.13353153738808215739894937211, 6.33738140739165496773939797422, 6.48706647654754191234602591050, 6.93837220696184295005558493422, 7.26307822358259254531316984862, 7.50155607756184202835593213827, 7.56538731061693665212905206945, 7.68155212738953520809810581499
