| L(s) = 1 | − 6·2-s − 9·3-s + 24·4-s − 6·5-s + 54·6-s + 6·7-s − 80·8-s + 54·9-s + 36·10-s − 45·11-s − 216·12-s + 39·13-s − 36·14-s + 54·15-s + 240·16-s + 72·17-s − 324·18-s − 3·19-s − 144·20-s − 54·21-s + 270·22-s − 117·23-s + 720·24-s + 12·25-s − 234·26-s − 270·27-s + 144·28-s + ⋯ |
| L(s) = 1 | − 2.12·2-s − 1.73·3-s + 3·4-s − 0.536·5-s + 3.67·6-s + 0.323·7-s − 3.53·8-s + 2·9-s + 1.13·10-s − 1.23·11-s − 5.19·12-s + 0.832·13-s − 0.687·14-s + 0.929·15-s + 15/4·16-s + 1.02·17-s − 4.24·18-s − 0.0362·19-s − 1.60·20-s − 0.561·21-s + 2.61·22-s − 1.06·23-s + 6.12·24-s + 0.0959·25-s − 1.76·26-s − 1.92·27-s + 0.971·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 44361864 ^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 44361864 ^{s/2} \, \Gamma_{\C}(s+3/2)^{3} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | $C_1$ | \( ( 1 + p T )^{3} \) |
| 3 | $C_1$ | \( ( 1 + p T )^{3} \) |
| 59 | $C_1$ | \( ( 1 - p T )^{3} \) |
| good | 5 | $S_4\times C_2$ | \( 1 + 6 T + 24 T^{2} - 1096 T^{3} + 24 p^{3} T^{4} + 6 p^{6} T^{5} + p^{9} T^{6} \) |
| 7 | $S_4\times C_2$ | \( 1 - 6 T + 963 T^{2} - 3705 T^{3} + 963 p^{3} T^{4} - 6 p^{6} T^{5} + p^{9} T^{6} \) |
| 11 | $S_4\times C_2$ | \( 1 + 45 T + 3060 T^{2} + 74653 T^{3} + 3060 p^{3} T^{4} + 45 p^{6} T^{5} + p^{9} T^{6} \) |
| 13 | $S_4\times C_2$ | \( 1 - 3 p T + 306 p T^{2} - 121203 T^{3} + 306 p^{4} T^{4} - 3 p^{7} T^{5} + p^{9} T^{6} \) |
| 17 | $S_4\times C_2$ | \( 1 - 72 T + 16347 T^{2} - 718441 T^{3} + 16347 p^{3} T^{4} - 72 p^{6} T^{5} + p^{9} T^{6} \) |
| 19 | $S_4\times C_2$ | \( 1 + 3 T + 15978 T^{2} + 110774 T^{3} + 15978 p^{3} T^{4} + 3 p^{6} T^{5} + p^{9} T^{6} \) |
| 23 | $S_4\times C_2$ | \( 1 + 117 T + 38946 T^{2} + 2830628 T^{3} + 38946 p^{3} T^{4} + 117 p^{6} T^{5} + p^{9} T^{6} \) |
| 29 | $S_4\times C_2$ | \( 1 + 3 T + 618 T^{2} + 2266784 T^{3} + 618 p^{3} T^{4} + 3 p^{6} T^{5} + p^{9} T^{6} \) |
| 31 | $S_4\times C_2$ | \( 1 - 39 T + 75078 T^{2} - 2042090 T^{3} + 75078 p^{3} T^{4} - 39 p^{6} T^{5} + p^{9} T^{6} \) |
| 37 | $S_4\times C_2$ | \( 1 + 24 T + 144783 T^{2} + 2393785 T^{3} + 144783 p^{3} T^{4} + 24 p^{6} T^{5} + p^{9} T^{6} \) |
| 41 | $S_4\times C_2$ | \( 1 + 504 T + 242355 T^{2} + 67793285 T^{3} + 242355 p^{3} T^{4} + 504 p^{6} T^{5} + p^{9} T^{6} \) |
| 43 | $S_4\times C_2$ | \( 1 - 201 T - 20844 T^{2} + 34377423 T^{3} - 20844 p^{3} T^{4} - 201 p^{6} T^{5} + p^{9} T^{6} \) |
| 47 | $S_4\times C_2$ | \( 1 + 663 T + 353562 T^{2} + 112422254 T^{3} + 353562 p^{3} T^{4} + 663 p^{6} T^{5} + p^{9} T^{6} \) |
| 53 | $S_4\times C_2$ | \( 1 + 1098 T + 684852 T^{2} + 298031580 T^{3} + 684852 p^{3} T^{4} + 1098 p^{6} T^{5} + p^{9} T^{6} \) |
| 61 | $S_4\times C_2$ | \( 1 + 243 T + 502188 T^{2} + 113179256 T^{3} + 502188 p^{3} T^{4} + 243 p^{6} T^{5} + p^{9} T^{6} \) |
| 67 | $S_4\times C_2$ | \( 1 + 330 T + 470982 T^{2} + 166720184 T^{3} + 470982 p^{3} T^{4} + 330 p^{6} T^{5} + p^{9} T^{6} \) |
| 71 | $S_4\times C_2$ | \( 1 + 2271 T + 38148 p T^{2} + 1987178235 T^{3} + 38148 p^{4} T^{4} + 2271 p^{6} T^{5} + p^{9} T^{6} \) |
| 73 | $S_4\times C_2$ | \( 1 + 381 T + 626718 T^{2} + 345144422 T^{3} + 626718 p^{3} T^{4} + 381 p^{6} T^{5} + p^{9} T^{6} \) |
| 79 | $S_4\times C_2$ | \( 1 + 1113 T + 1537488 T^{2} + 938879449 T^{3} + 1537488 p^{3} T^{4} + 1113 p^{6} T^{5} + p^{9} T^{6} \) |
| 83 | $S_4\times C_2$ | \( 1 + 2262 T + 2959101 T^{2} + 2566898023 T^{3} + 2959101 p^{3} T^{4} + 2262 p^{6} T^{5} + p^{9} T^{6} \) |
| 89 | $S_4\times C_2$ | \( 1 + 2055 T + 3490512 T^{2} + 3198012690 T^{3} + 3490512 p^{3} T^{4} + 2055 p^{6} T^{5} + p^{9} T^{6} \) |
| 97 | $S_4\times C_2$ | \( 1 + 18 T + 989952 T^{2} - 720061856 T^{3} + 989952 p^{3} T^{4} + 18 p^{6} T^{5} + p^{9} T^{6} \) |
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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
−10.15296265902213823520887495233, −9.999600477631872130087203076931, −9.891557482851237944668454270425, −9.601691975983037833601557951312, −8.977315425093460074278192460152, −8.642227121082432130092726885437, −8.309351346066967166235993015522, −8.109420009485605280298938348940, −7.71165494341573248409225539024, −7.68372828546538063354967056312, −7.07395325313274651022145617804, −6.80800228371007767715727959265, −6.60525170059651755102798118454, −5.82852154365470589535939960788, −5.82191687295601332768694171268, −5.80396657367384365959088029077, −4.95246018511125277316814271889, −4.59779798168936298448023389474, −4.40736542333274059748988201433, −3.41550356481110216697982349833, −3.21713143614586783637356097973, −2.78673631217151103305427024880, −1.74518756531830164974551960228, −1.43820514846492083335553541529, −1.33581459878275752351912676108, 0, 0, 0,
1.33581459878275752351912676108, 1.43820514846492083335553541529, 1.74518756531830164974551960228, 2.78673631217151103305427024880, 3.21713143614586783637356097973, 3.41550356481110216697982349833, 4.40736542333274059748988201433, 4.59779798168936298448023389474, 4.95246018511125277316814271889, 5.80396657367384365959088029077, 5.82191687295601332768694171268, 5.82852154365470589535939960788, 6.60525170059651755102798118454, 6.80800228371007767715727959265, 7.07395325313274651022145617804, 7.68372828546538063354967056312, 7.71165494341573248409225539024, 8.109420009485605280298938348940, 8.309351346066967166235993015522, 8.642227121082432130092726885437, 8.977315425093460074278192460152, 9.601691975983037833601557951312, 9.891557482851237944668454270425, 9.999600477631872130087203076931, 10.15296265902213823520887495233
