| L(s) = 1 | + 24·3-s + 333·9-s + 1.29e3·11-s + 484·13-s − 2.59e3·23-s − 86·25-s + 2.16e3·27-s + 3.11e4·33-s + 2.41e4·37-s + 1.16e4·39-s − 2.59e4·47-s + 1.53e3·49-s + 1.68e4·59-s + 5.15e4·61-s − 6.22e4·69-s − 1.11e5·71-s + 5.20e4·73-s − 2.06e3·75-s − 2.90e4·81-s − 1.56e5·83-s + 2.06e5·97-s + 4.31e5·99-s + 4.27e4·107-s − 1.17e5·109-s + 5.78e5·111-s + 1.61e5·117-s + 9.37e5·121-s + ⋯ |
| L(s) = 1 | + 1.53·3-s + 1.37·9-s + 3.22·11-s + 0.794·13-s − 1.02·23-s − 0.0275·25-s + 0.570·27-s + 4.97·33-s + 2.89·37-s + 1.22·39-s − 1.71·47-s + 0.0915·49-s + 0.630·59-s + 1.77·61-s − 1.57·69-s − 2.62·71-s + 1.14·73-s − 0.0423·75-s − 0.492·81-s − 2.49·83-s + 2.22·97-s + 4.42·99-s + 0.361·107-s − 0.949·109-s + 4.45·111-s + 1.08·117-s + 5.82·121-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 36864 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 36864 ^{s/2} \, \Gamma_{\C}(s+5/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(7.960094250\) |
| \(L(\frac12)\) |
\(\approx\) |
\(7.960094250\) |
| \(L(\frac{7}{2})\) |
|
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 | | \( 1 \) |
| 3 | $C_2$ | \( 1 - 8 p T + p^{5} T^{2} \) |
| good | 5 | $C_2^2$ | \( 1 + 86 T^{2} + p^{10} T^{4} \) |
| 7 | $C_2^2$ | \( 1 - 1538 T^{2} + p^{10} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - 648 T + p^{5} T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 242 T + p^{5} T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 - 2738338 T^{2} + p^{10} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 3380474 T^{2} + p^{10} T^{4} \) |
| 23 | $C_2$ | \( ( 1 + 1296 T + p^{5} T^{2} )^{2} \) |
| 29 | $C_2^2$ | \( 1 - 37062298 T^{2} + p^{10} T^{4} \) |
| 31 | $C_2^2$ | \( 1 - 45678866 T^{2} + p^{10} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - 12058 T + p^{5} T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - 12512146 T^{2} + p^{10} T^{4} \) |
| 43 | $C_2^2$ | \( 1 + 33190390 T^{2} + p^{10} T^{4} \) |
| 47 | $C_2$ | \( ( 1 + 12960 T + p^{5} T^{2} )^{2} \) |
| 53 | $C_2^2$ | \( 1 - 108251530 T^{2} + p^{10} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - 8424 T + p^{5} T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 25762 T + p^{5} T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 2596035290 T^{2} + p^{10} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 55728 T + p^{5} T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 26026 T + p^{5} T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 - 5786874674 T^{2} + p^{10} T^{4} \) |
| 83 | $C_2$ | \( ( 1 + 78408 T + p^{5} T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - 4075828594 T^{2} + p^{10} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 103090 T + p^{5} T^{2} )^{2} \) |
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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
−11.70195792194249950649138961505, −11.54940421548692553848546479804, −11.13980508107359595599901028338, −9.992710033388610245655190502357, −9.866194319862985254624453985815, −9.157887117672294528940696415866, −9.093134977812673500917715571882, −8.296900105757629231271298032361, −8.182251404421079821546200406183, −7.27699135984698108832791716690, −6.81339465976824762791958469538, −6.19808371796912536384200007598, −5.87110770408895107730628401812, −4.42965363099521917612615083294, −4.16756123793512225662955790163, −3.65161404112628769626621923557, −3.06846339842153010034966697622, −2.08752641392113306644893999713, −1.47339594686954135067058355373, −0.877705713219135445031799578805,
0.877705713219135445031799578805, 1.47339594686954135067058355373, 2.08752641392113306644893999713, 3.06846339842153010034966697622, 3.65161404112628769626621923557, 4.16756123793512225662955790163, 4.42965363099521917612615083294, 5.87110770408895107730628401812, 6.19808371796912536384200007598, 6.81339465976824762791958469538, 7.27699135984698108832791716690, 8.182251404421079821546200406183, 8.296900105757629231271298032361, 9.093134977812673500917715571882, 9.157887117672294528940696415866, 9.866194319862985254624453985815, 9.992710033388610245655190502357, 11.13980508107359595599901028338, 11.54940421548692553848546479804, 11.70195792194249950649138961505
