| L(s) = 1 | − 19·4-s + 190·7-s − 1.17e3·13-s − 3.73e3·16-s + 1.66e4·19-s − 2.75e4·25-s − 3.61e3·28-s − 7.02e4·31-s − 1.24e5·37-s − 2.33e4·43-s − 2.08e5·49-s + 2.23e4·52-s + 4.37e5·61-s + 1.48e5·64-s + 2.56e5·67-s − 9.39e5·73-s − 3.15e5·76-s − 4.51e5·79-s − 2.23e5·91-s − 2.70e6·97-s + 5.23e5·100-s + 3.50e6·103-s + 8.70e5·109-s − 7.09e5·112-s + 1.46e5·121-s + 1.33e6·124-s + 127-s + ⋯ |
| L(s) = 1 | − 0.296·4-s + 0.553·7-s − 0.536·13-s − 0.911·16-s + 2.42·19-s − 1.76·25-s − 0.164·28-s − 2.35·31-s − 2.45·37-s − 0.293·43-s − 1.76·49-s + 0.159·52-s + 1.92·61-s + 0.567·64-s + 0.852·67-s − 2.41·73-s − 0.719·76-s − 0.915·79-s − 0.297·91-s − 2.96·97-s + 0.523·100-s + 3.20·103-s + 0.672·109-s − 0.505·112-s + 0.0828·121-s + 0.699·124-s + 1.34·133-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 59049 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 59049 ^{s/2} \, \Gamma_{\C}(s+3)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(0.1328869057\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1328869057\) |
| \(L(4)\) |
|
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 | | \( 1 \) |
| good | 2 | $C_2^2$ | \( 1 + 19 T^{2} + p^{12} T^{4} \) |
| 5 | $C_2^2$ | \( 1 + 1102 p^{2} T^{2} + p^{12} T^{4} \) |
| 7 | $C_2$ | \( ( 1 - 95 T + p^{6} T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( 1 - 146834 T^{2} + p^{12} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + 589 T + p^{6} T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 - 40633490 T^{2} + p^{12} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 8315 T + p^{6} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 191358386 T^{2} + p^{12} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 214187570 T^{2} + p^{12} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + 35113 T + p^{6} T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + 62245 T + p^{6} T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - 6470303282 T^{2} + p^{12} T^{4} \) |
| 43 | $C_2$ | \( ( 1 + 11677 T + p^{6} T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 1941471170 T^{2} + p^{12} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 23192879470 T^{2} + p^{12} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 83911908194 T^{2} + p^{12} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 218906 T + p^{6} T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 128186 T + p^{6} T^{2} )^{2} \) |
| 71 | $C_2^2$ | \( 1 - 235537551650 T^{2} + p^{12} T^{4} \) |
| 73 | $C_2$ | \( ( 1 + 469726 T + p^{6} T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + 225625 T + p^{6} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 608903147330 T^{2} + p^{12} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 494599957490 T^{2} + p^{12} T^{4} \) |
| 97 | $C_2$ | \( ( 1 + 1354081 T + p^{6} T^{2} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−11.29598439279144254468752396351, −11.05164839414897336390861576460, −10.02705474253363666275218941072, −9.920884014213472524371271126578, −9.459246782410401017924084312939, −8.761539169348307111381821178637, −8.541222284481593164852241464873, −7.66904485535851729363833877948, −7.33374972664173769949725509306, −7.07257903005486957744951813791, −6.17869986946553509995688780931, −5.37438744074638951094592881402, −5.28097885909244907778334544101, −4.66676418666084098136127782219, −3.66038956557283824295735519564, −3.56210907683469376120196238900, −2.52473382745742673773327979263, −1.77921882663865305158960274526, −1.33048051254519677461058613358, −0.092874370352313469462097691937,
0.092874370352313469462097691937, 1.33048051254519677461058613358, 1.77921882663865305158960274526, 2.52473382745742673773327979263, 3.56210907683469376120196238900, 3.66038956557283824295735519564, 4.66676418666084098136127782219, 5.28097885909244907778334544101, 5.37438744074638951094592881402, 6.17869986946553509995688780931, 7.07257903005486957744951813791, 7.33374972664173769949725509306, 7.66904485535851729363833877948, 8.541222284481593164852241464873, 8.761539169348307111381821178637, 9.459246782410401017924084312939, 9.920884014213472524371271126578, 10.02705474253363666275218941072, 11.05164839414897336390861576460, 11.29598439279144254468752396351
