| L(s) = 1 | − 148·4-s − 1.11e3·7-s − 1.73e4·13-s + 5.52e3·16-s − 6.49e4·19-s − 3.14e4·25-s + 1.65e5·28-s + 4.59e5·31-s − 1.08e6·37-s − 9.30e5·43-s − 7.09e5·49-s + 2.56e6·52-s − 2.75e5·61-s + 1.60e6·64-s − 6.28e5·67-s + 5.33e6·73-s + 9.60e6·76-s + 2.20e6·79-s + 1.93e7·91-s − 5.95e6·97-s + 4.64e6·100-s − 1.17e7·103-s + 1.78e7·109-s − 6.17e6·112-s − 1.67e7·121-s − 6.80e7·124-s + 127-s + ⋯ |
| L(s) = 1 | − 1.15·4-s − 1.23·7-s − 2.18·13-s + 0.336·16-s − 2.17·19-s − 0.401·25-s + 1.42·28-s + 2.77·31-s − 3.51·37-s − 1.78·43-s − 0.861·49-s + 2.53·52-s − 0.155·61-s + 0.766·64-s − 0.255·67-s + 1.60·73-s + 2.51·76-s + 0.502·79-s + 2.69·91-s − 0.662·97-s + 0.464·100-s − 1.05·103-s + 1.32·109-s − 0.415·112-s − 0.857·121-s − 3.20·124-s + 2.67·133-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s+7/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{9}{2})\) |
|
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 | | \( 1 \) |
| good | 2 | $C_2^2$ | \( 1 + 37 p^{2} T^{2} + p^{14} T^{4} \) |
| 5 | $C_2^2$ | \( 1 + 31402 T^{2} + p^{14} T^{4} \) |
| 7 | $C_2$ | \( ( 1 + 559 T + p^{7} T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( 1 + 16713814 T^{2} + p^{14} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + 667 p T + p^{7} T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 + 189231154 T^{2} + p^{14} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + 32461 T + p^{7} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 + 18081694 T^{2} + p^{14} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 9599936170 T^{2} + p^{14} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 229892 T + p^{7} T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + 541177 T + p^{7} T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 + 264543016114 T^{2} + p^{14} T^{4} \) |
| 43 | $C_2$ | \( ( 1 + 465112 T + p^{7} T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 + 323393423854 T^{2} + p^{14} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 1296296189242 T^{2} + p^{14} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 4360664458486 T^{2} + p^{14} T^{4} \) |
| 61 | $C_2$ | \( ( 1 + 137773 T + p^{7} T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 + 314041 T + p^{7} T^{2} )^{2} \) |
| 71 | $C_2^2$ | \( 1 + 10295330331310 T^{2} + p^{14} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 36569 p T + p^{7} T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 1101815 T + p^{7} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 + 17317361610166 T^{2} + p^{14} T^{4} \) |
| 89 | $C_2^2$ | \( 1 + 77662517197858 T^{2} + p^{14} T^{4} \) |
| 97 | $C_2$ | \( ( 1 + 2979379 T + p^{7} T^{2} )^{2} \) |
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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
−15.30809859411365159597219379717, −14.93324438020422445147537383513, −14.03169805171405428311262134190, −13.62928834236558798218828006344, −12.92765914721750209441987748146, −12.33174935363876988163628468021, −11.98033299180455238610726822627, −10.62339328422532919484579588179, −9.855449691442024408834874111711, −9.777998979349594916686192167271, −8.687511773366402681957283490438, −8.220658638036765700153006701231, −6.90743720729900396756342078891, −6.46373836221569636960281246080, −5.05739494140163633204737767854, −4.54985839463128382303112830726, −3.38815430353768904665244101178, −2.21375110206006730763230573384, 0, 0,
2.21375110206006730763230573384, 3.38815430353768904665244101178, 4.54985839463128382303112830726, 5.05739494140163633204737767854, 6.46373836221569636960281246080, 6.90743720729900396756342078891, 8.220658638036765700153006701231, 8.687511773366402681957283490438, 9.777998979349594916686192167271, 9.855449691442024408834874111711, 10.62339328422532919484579588179, 11.98033299180455238610726822627, 12.33174935363876988163628468021, 12.92765914721750209441987748146, 13.62928834236558798218828006344, 14.03169805171405428311262134190, 14.93324438020422445147537383513, 15.30809859411365159597219379717
