| L(s) = 1 | − 6·2-s + 128·4-s − 390·5-s + 64·7-s − 2.08e3·8-s + 2.34e3·10-s + 948·11-s + 5.09e3·13-s − 384·14-s + 1.25e4·16-s + 5.67e4·17-s − 1.72e4·19-s − 4.99e4·20-s − 5.68e3·22-s + 1.52e4·23-s + 7.81e4·25-s − 3.05e4·26-s + 8.19e3·28-s − 3.65e4·29-s + 2.76e5·31-s − 2.67e5·32-s − 3.40e5·34-s − 2.49e4·35-s + 5.37e5·37-s + 1.03e5·38-s + 8.14e5·40-s + 6.29e5·41-s + ⋯ |
| L(s) = 1 | − 0.530·2-s + 4-s − 1.39·5-s + 0.0705·7-s − 1.44·8-s + 0.739·10-s + 0.214·11-s + 0.643·13-s − 0.0374·14-s + 0.764·16-s + 2.80·17-s − 0.576·19-s − 1.39·20-s − 0.113·22-s + 0.262·23-s + 25-s − 0.341·26-s + 0.0705·28-s − 0.277·29-s + 1.66·31-s − 1.44·32-s − 1.48·34-s − 0.0984·35-s + 1.74·37-s + 0.305·38-s + 2.01·40-s + 1.42·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6561 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6561 ^{s/2} \, \Gamma_{\C}(s+7/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(2.023948114\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.023948114\) |
| \(L(\frac{9}{2})\) |
|
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 | $C_2^2$ | \( 1 + 3 p T - 23 p^{2} T^{2} + 3 p^{8} T^{3} + p^{14} T^{4} \) |
| 5 | $C_2^2$ | \( 1 + 78 p T + 2959 p^{2} T^{2} + 78 p^{8} T^{3} + p^{14} T^{4} \) |
| 7 | $C_2^2$ | \( 1 - 64 T - 819447 T^{2} - 64 p^{7} T^{3} + p^{14} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 948 T - 18588467 T^{2} - 948 p^{7} T^{3} + p^{14} T^{4} \) |
| 13 | $C_2^2$ | \( 1 - 5098 T - 36758913 T^{2} - 5098 p^{7} T^{3} + p^{14} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - 28386 T + p^{7} T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + 8620 T + p^{7} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 15288 T - 3171102503 T^{2} - 15288 p^{7} T^{3} + p^{14} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 36510 T - 15916896209 T^{2} + 36510 p^{7} T^{3} + p^{14} T^{4} \) |
| 31 | $C_2^2$ | \( 1 - 276808 T + 49110054753 T^{2} - 276808 p^{7} T^{3} + p^{14} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - 268526 T + p^{7} T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - 629718 T + 201790485643 T^{2} - 629718 p^{7} T^{3} + p^{14} T^{4} \) |
| 43 | $C_2^2$ | \( 1 + 685772 T + 198464624877 T^{2} + 685772 p^{7} T^{3} + p^{14} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 583296 T - 166388896847 T^{2} + 583296 p^{7} T^{3} + p^{14} T^{4} \) |
| 53 | $C_2$ | \( ( 1 + 428058 T + p^{7} T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 + 1306380 T - 782022780419 T^{2} + 1306380 p^{7} T^{3} + p^{14} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 300662 T - 3052345197777 T^{2} + 300662 p^{7} T^{3} + p^{14} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 507244 T - 5803415129787 T^{2} - 507244 p^{7} T^{3} + p^{14} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 5560632 T + p^{7} T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 1369082 T + p^{7} T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 - 6913720 T + 28595615252241 T^{2} - 6913720 p^{7} T^{3} + p^{14} T^{4} \) |
| 83 | $C_2^2$ | \( 1 - 4376748 T - 7980127934123 T^{2} - 4376748 p^{7} T^{3} + p^{14} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 8528310 T + p^{7} T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 8826814 T - 2885639087517 T^{2} - 8826814 p^{7} T^{3} + p^{14} T^{4} \) |
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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
−12.86902761873667888823174092314, −12.30825338308689239542730527839, −12.17737686359930592672743205534, −11.38655811870446620277468566904, −11.25888760112186086294701085431, −10.57237691272699703989176721023, −9.712181534914462576202066887373, −9.457432660188562358321096651550, −8.400121828715746540671205924802, −8.040272922280292031462158884487, −7.70926737391755823941387744782, −6.85842242960574059753073927991, −6.26912618919320293550261903277, −5.67727075398821802535828907984, −4.67396488317528629529747915858, −3.61870046068402110000549872473, −3.29656150328313478376054519022, −2.40589340426000954177657622317, −1.11210981578626505466538723934, −0.62040112999915299392346281810,
0.62040112999915299392346281810, 1.11210981578626505466538723934, 2.40589340426000954177657622317, 3.29656150328313478376054519022, 3.61870046068402110000549872473, 4.67396488317528629529747915858, 5.67727075398821802535828907984, 6.26912618919320293550261903277, 6.85842242960574059753073927991, 7.70926737391755823941387744782, 8.040272922280292031462158884487, 8.400121828715746540671205924802, 9.457432660188562358321096651550, 9.712181534914462576202066887373, 10.57237691272699703989176721023, 11.25888760112186086294701085431, 11.38655811870446620277468566904, 12.17737686359930592672743205534, 12.30825338308689239542730527839, 12.86902761873667888823174092314
