| L(s) = 1 | − 2-s − 3-s − 2·4-s − 7·5-s + 6-s + 2·7-s + 2·8-s − 3·9-s + 7·10-s − 10·11-s + 2·12-s + 3·13-s − 2·14-s + 7·15-s + 16-s + 17-s + 3·18-s − 5·19-s + 14·20-s − 2·21-s + 10·22-s + 4·23-s − 2·24-s + 21·25-s − 3·26-s + 2·27-s − 4·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s − 4-s − 3.13·5-s + 0.408·6-s + 0.755·7-s + 0.707·8-s − 9-s + 2.21·10-s − 3.01·11-s + 0.577·12-s + 0.832·13-s − 0.534·14-s + 1.80·15-s + 1/4·16-s + 0.242·17-s + 0.707·18-s − 1.14·19-s + 3.13·20-s − 0.436·21-s + 2.13·22-s + 0.834·23-s − 0.408·24-s + 21/5·25-s − 0.588·26-s + 0.384·27-s − 0.755·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(13^{3} \cdot 103^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(13^{3} \cdot 103^{3}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 13 | $C_1$ | \( ( 1 - T )^{3} \) |
| 103 | $C_1$ | \( ( 1 - T )^{3} \) |
| good | 2 | $S_4\times C_2$ | \( 1 + T + 3 T^{2} + 3 T^{3} + 3 p T^{4} + p^{2} T^{5} + p^{3} T^{6} \) |
| 3 | $S_4\times C_2$ | \( 1 + T + 4 T^{2} + 5 T^{3} + 4 p T^{4} + p^{2} T^{5} + p^{3} T^{6} \) |
| 5 | $S_4\times C_2$ | \( 1 + 7 T + 28 T^{2} + 3 p^{2} T^{3} + 28 p T^{4} + 7 p^{2} T^{5} + p^{3} T^{6} \) |
| 7 | $S_4\times C_2$ | \( 1 - 2 T + 17 T^{2} - 24 T^{3} + 17 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 11 | $S_4\times C_2$ | \( 1 + 10 T + 57 T^{2} + 216 T^{3} + 57 p T^{4} + 10 p^{2} T^{5} + p^{3} T^{6} \) |
| 17 | $S_4\times C_2$ | \( 1 - T + 30 T^{2} - 21 T^{3} + 30 p T^{4} - p^{2} T^{5} + p^{3} T^{6} \) |
| 19 | $S_4\times C_2$ | \( 1 + 5 T + 44 T^{2} + 123 T^{3} + 44 p T^{4} + 5 p^{2} T^{5} + p^{3} T^{6} \) |
| 23 | $S_4\times C_2$ | \( 1 - 4 T - 11 T^{2} + 216 T^{3} - 11 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 29 | $S_4\times C_2$ | \( 1 + 17 T + 178 T^{2} + 1137 T^{3} + 178 p T^{4} + 17 p^{2} T^{5} + p^{3} T^{6} \) |
| 31 | $S_4\times C_2$ | \( 1 + 8 T + 61 T^{2} + 224 T^{3} + 61 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 37 | $S_4\times C_2$ | \( 1 - 3 T + 80 T^{2} - 251 T^{3} + 80 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 41 | $S_4\times C_2$ | \( 1 + 8 T + 107 T^{2} + 496 T^{3} + 107 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 43 | $S_4\times C_2$ | \( 1 - 8 T + 137 T^{2} - 672 T^{3} + 137 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 47 | $S_4\times C_2$ | \( 1 + 4 T + 69 T^{2} + 108 T^{3} + 69 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 53 | $S_4\times C_2$ | \( 1 + 12 T + 179 T^{2} + 1172 T^{3} + 179 p T^{4} + 12 p^{2} T^{5} + p^{3} T^{6} \) |
| 59 | $S_4\times C_2$ | \( 1 + 11 T + 98 T^{2} + 475 T^{3} + 98 p T^{4} + 11 p^{2} T^{5} + p^{3} T^{6} \) |
| 61 | $D_{6}$ | \( 1 - 9 T + 162 T^{2} - 1001 T^{3} + 162 p T^{4} - 9 p^{2} T^{5} + p^{3} T^{6} \) |
| 67 | $S_4\times C_2$ | \( 1 - 8 T + 169 T^{2} - 944 T^{3} + 169 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 71 | $S_4\times C_2$ | \( 1 - 10 T + 97 T^{2} - 324 T^{3} + 97 p T^{4} - 10 p^{2} T^{5} + p^{3} T^{6} \) |
| 73 | $S_4\times C_2$ | \( 1 - 9 T - 16 T^{2} + 1075 T^{3} - 16 p T^{4} - 9 p^{2} T^{5} + p^{3} T^{6} \) |
| 79 | $S_4\times C_2$ | \( 1 - 6 T + 129 T^{2} - 732 T^{3} + 129 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 83 | $S_4\times C_2$ | \( 1 + 11 T + 230 T^{2} + 1831 T^{3} + 230 p T^{4} + 11 p^{2} T^{5} + p^{3} T^{6} \) |
| 89 | $S_4\times C_2$ | \( 1 + T + 108 T^{2} + 345 T^{3} + 108 p T^{4} + p^{2} T^{5} + p^{3} T^{6} \) |
| 97 | $S_4\times C_2$ | \( 1 - 22 T + 415 T^{2} - 4468 T^{3} + 415 p T^{4} - 22 p^{2} T^{5} + p^{3} T^{6} \) |
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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
−8.944453343451633870504240753696, −8.378618604513700836006543010479, −8.296662495564504086450322323302, −8.287515655668730613103611923475, −7.86234610952019325963245892582, −7.65020689832343175860820575071, −7.61455349790650528441585672606, −7.42134567138561265504589502246, −6.96607345943087411242264493281, −6.34831552855298886747637381296, −6.28657038268814874053185084767, −5.70164607445409131919060996190, −5.49073007500430062813703324406, −5.06858161784997611525524788325, −4.93597821966455124410520437796, −4.93337076943410095699950113503, −4.23890140503091480871358111845, −3.92945465276919834104895503363, −3.84616131678550058788098321201, −3.37123226052790036756153596399, −3.17737415089227285402559236251, −2.73700805129373733884572911404, −2.05931018707618849463116477124, −1.78878245489352560395458767859, −0.73749877297560804180048932186, 0, 0, 0,
0.73749877297560804180048932186, 1.78878245489352560395458767859, 2.05931018707618849463116477124, 2.73700805129373733884572911404, 3.17737415089227285402559236251, 3.37123226052790036756153596399, 3.84616131678550058788098321201, 3.92945465276919834104895503363, 4.23890140503091480871358111845, 4.93337076943410095699950113503, 4.93597821966455124410520437796, 5.06858161784997611525524788325, 5.49073007500430062813703324406, 5.70164607445409131919060996190, 6.28657038268814874053185084767, 6.34831552855298886747637381296, 6.96607345943087411242264493281, 7.42134567138561265504589502246, 7.61455349790650528441585672606, 7.65020689832343175860820575071, 7.86234610952019325963245892582, 8.287515655668730613103611923475, 8.296662495564504086450322323302, 8.378618604513700836006543010479, 8.944453343451633870504240753696
