| L(s) = 1 | + 2-s − 9·3-s − 14·4-s + 15·5-s − 9·6-s − 16·7-s − 18·8-s + 54·9-s + 15·10-s − 33·11-s + 126·12-s − 42·13-s − 16·14-s − 135·15-s + 83·16-s − 34·17-s + 54·18-s − 280·19-s − 210·20-s + 144·21-s − 33·22-s − 112·23-s + 162·24-s + 150·25-s − 42·26-s − 270·27-s + 224·28-s + ⋯ |
| L(s) = 1 | + 0.353·2-s − 1.73·3-s − 7/4·4-s + 1.34·5-s − 0.612·6-s − 0.863·7-s − 0.795·8-s + 2·9-s + 0.474·10-s − 0.904·11-s + 3.03·12-s − 0.896·13-s − 0.305·14-s − 2.32·15-s + 1.29·16-s − 0.485·17-s + 0.707·18-s − 3.38·19-s − 2.34·20-s + 1.49·21-s − 0.319·22-s − 1.01·23-s + 1.37·24-s + 6/5·25-s − 0.316·26-s − 1.92·27-s + 1.51·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4492125 ^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4492125 ^{s/2} \, \Gamma_{\C}(s+3/2)^{3} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | $C_1$ | \( ( 1 - p T )^{3} \) |
| 11 | $C_1$ | \( ( 1 + p T )^{3} \) |
| good | 2 | $S_4\times C_2$ | \( 1 - T + 15 T^{2} - 11 T^{3} + 15 p^{3} T^{4} - p^{6} T^{5} + p^{9} T^{6} \) |
| 7 | $S_4\times C_2$ | \( 1 + 16 T + 277 T^{2} + 5168 T^{3} + 277 p^{3} T^{4} + 16 p^{6} T^{5} + p^{9} T^{6} \) |
| 13 | $S_4\times C_2$ | \( 1 + 42 T + 1363 T^{2} + 47132 T^{3} + 1363 p^{3} T^{4} + 42 p^{6} T^{5} + p^{9} T^{6} \) |
| 17 | $S_4\times C_2$ | \( 1 + 2 p T + 10079 T^{2} + 155020 T^{3} + 10079 p^{3} T^{4} + 2 p^{7} T^{5} + p^{9} T^{6} \) |
| 19 | $S_4\times C_2$ | \( 1 + 280 T + 39201 T^{2} + 3743984 T^{3} + 39201 p^{3} T^{4} + 280 p^{6} T^{5} + p^{9} T^{6} \) |
| 23 | $S_4\times C_2$ | \( 1 + 112 T + 19477 T^{2} + 809120 T^{3} + 19477 p^{3} T^{4} + 112 p^{6} T^{5} + p^{9} T^{6} \) |
| 29 | $S_4\times C_2$ | \( 1 + 10 p T + 46667 T^{2} + 4894124 T^{3} + 46667 p^{3} T^{4} + 10 p^{7} T^{5} + p^{9} T^{6} \) |
| 31 | $S_4\times C_2$ | \( 1 + 392 T + 121533 T^{2} + 23039344 T^{3} + 121533 p^{3} T^{4} + 392 p^{6} T^{5} + p^{9} T^{6} \) |
| 37 | $S_4\times C_2$ | \( 1 - 570 T + 212899 T^{2} - 56704796 T^{3} + 212899 p^{3} T^{4} - 570 p^{6} T^{5} + p^{9} T^{6} \) |
| 41 | $S_4\times C_2$ | \( 1 + 662 T + 150855 T^{2} + 22689620 T^{3} + 150855 p^{3} T^{4} + 662 p^{6} T^{5} + p^{9} T^{6} \) |
| 43 | $S_4\times C_2$ | \( 1 + 68 T + 11145 T^{2} - 9678104 T^{3} + 11145 p^{3} T^{4} + 68 p^{6} T^{5} + p^{9} T^{6} \) |
| 47 | $S_4\times C_2$ | \( 1 - 264 T + 146893 T^{2} - 68940144 T^{3} + 146893 p^{3} T^{4} - 264 p^{6} T^{5} + p^{9} T^{6} \) |
| 53 | $S_4\times C_2$ | \( 1 + 94 T + 388819 T^{2} + 25584884 T^{3} + 388819 p^{3} T^{4} + 94 p^{6} T^{5} + p^{9} T^{6} \) |
| 59 | $S_4\times C_2$ | \( 1 + 612 T + 191353 T^{2} + 89255576 T^{3} + 191353 p^{3} T^{4} + 612 p^{6} T^{5} + p^{9} T^{6} \) |
| 61 | $S_4\times C_2$ | \( 1 + 582 T + 700987 T^{2} + 242850884 T^{3} + 700987 p^{3} T^{4} + 582 p^{6} T^{5} + p^{9} T^{6} \) |
| 67 | $S_4\times C_2$ | \( 1 - 940 T + 512801 T^{2} - 170936200 T^{3} + 512801 p^{3} T^{4} - 940 p^{6} T^{5} + p^{9} T^{6} \) |
| 71 | $S_4\times C_2$ | \( 1 + 1616 T + 1744933 T^{2} + 1196967136 T^{3} + 1744933 p^{3} T^{4} + 1616 p^{6} T^{5} + p^{9} T^{6} \) |
| 73 | $S_4\times C_2$ | \( 1 - 738 T + 1335919 T^{2} - 586235356 T^{3} + 1335919 p^{3} T^{4} - 738 p^{6} T^{5} + p^{9} T^{6} \) |
| 79 | $S_4\times C_2$ | \( 1 - 124 T + 1293245 T^{2} - 95402344 T^{3} + 1293245 p^{3} T^{4} - 124 p^{6} T^{5} + p^{9} T^{6} \) |
| 83 | $S_4\times C_2$ | \( 1 + 1232 T + 2008521 T^{2} + 1393226768 T^{3} + 2008521 p^{3} T^{4} + 1232 p^{6} T^{5} + p^{9} T^{6} \) |
| 89 | $S_4\times C_2$ | \( 1 - 838 T + 1110375 T^{2} - 349581812 T^{3} + 1110375 p^{3} T^{4} - 838 p^{6} T^{5} + p^{9} T^{6} \) |
| 97 | $S_4\times C_2$ | \( 1 + 90 T + 441679 T^{2} - 759889556 T^{3} + 441679 p^{3} T^{4} + 90 p^{6} T^{5} + p^{9} T^{6} \) |
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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
−11.50139661980146337367212838679, −11.01984044534980877778517120185, −10.78788034835486251756389641174, −10.69250371373445468294269193823, −9.974342647212632823202518823828, −9.884227708582963514601395771885, −9.802339055074831377516236777900, −9.102814911172772686951739322571, −8.930664885777321707573469584434, −8.759653361251736844892746679812, −7.87288171642494111428089769999, −7.79221230379473379016271373658, −7.10489995270158592949818249408, −6.58766313599375996437627400655, −6.28360587873121776552872967025, −6.13304821742037061415423369891, −5.48175619802852043669215629708, −5.40295365547241660136231461319, −4.92423063742327970705801464112, −4.51229730165067202985344612662, −3.97769407883411266935685157409, −3.93782620886573547364832061283, −2.78091319458968493639424533927, −2.11135438955377781157363479411, −1.66347134938011861186152169094, 0, 0, 0,
1.66347134938011861186152169094, 2.11135438955377781157363479411, 2.78091319458968493639424533927, 3.93782620886573547364832061283, 3.97769407883411266935685157409, 4.51229730165067202985344612662, 4.92423063742327970705801464112, 5.40295365547241660136231461319, 5.48175619802852043669215629708, 6.13304821742037061415423369891, 6.28360587873121776552872967025, 6.58766313599375996437627400655, 7.10489995270158592949818249408, 7.79221230379473379016271373658, 7.87288171642494111428089769999, 8.759653361251736844892746679812, 8.930664885777321707573469584434, 9.102814911172772686951739322571, 9.802339055074831377516236777900, 9.884227708582963514601395771885, 9.974342647212632823202518823828, 10.69250371373445468294269193823, 10.78788034835486251756389641174, 11.01984044534980877778517120185, 11.50139661980146337367212838679
