| L(s) = 1 | − 8·3-s + 16·4-s + 18·5-s + 37·9-s − 128·12-s − 144·15-s + 192·16-s + 288·20-s + 216·23-s + 199·25-s − 80·27-s + 680·31-s + 592·36-s + 666·45-s − 72·47-s − 1.53e3·48-s − 686·49-s − 1.47e3·53-s − 2.30e3·60-s + 2.04e3·64-s − 1.72e3·69-s − 1.59e3·75-s + 3.45e3·80-s − 359·81-s + 3.45e3·92-s − 5.44e3·93-s + 3.18e3·100-s + ⋯ |
| L(s) = 1 | − 1.53·3-s + 2·4-s + 1.60·5-s + 1.37·9-s − 3.07·12-s − 2.47·15-s + 3·16-s + 3.21·20-s + 1.95·23-s + 1.59·25-s − 0.570·27-s + 3.93·31-s + 2.74·36-s + 2.20·45-s − 0.223·47-s − 4.61·48-s − 2·49-s − 3.82·53-s − 4.95·60-s + 4·64-s − 3.01·69-s − 2.45·75-s + 4.82·80-s − 0.492·81-s + 3.91·92-s − 6.06·93-s + 3.18·100-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 27225 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 27225 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(3.665353846\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.665353846\) |
| \(L(\frac{5}{2})\) |
|
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 | $C_2$ | \( 1 + 8 T + p^{3} T^{2} \) |
| 5 | $C_2$ | \( 1 - 18 T + p^{3} T^{2} \) |
| 11 | $C_2$ | \( 1 + p^{3} T^{2} \) |
| good | 2 | $C_2$ | \( ( 1 - p^{3} T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 + p^{3} T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 + p^{3} T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - p^{3} T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - p^{3} T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - 108 T + p^{3} T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + p^{3} T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 340 T + p^{3} T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 434 T + p^{3} T^{2} )( 1 + 434 T + p^{3} T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 + p^{3} T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + p^{3} T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 + 36 T + p^{3} T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 + 738 T + p^{3} T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 720 T + p^{3} T^{2} )( 1 + 720 T + p^{3} T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - p^{3} T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 416 T + p^{3} T^{2} )( 1 + 416 T + p^{3} T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 - 612 T + p^{3} T^{2} )( 1 + 612 T + p^{3} T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 + p^{3} T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - p^{3} T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - p^{3} T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 1674 T + p^{3} T^{2} )( 1 + 1674 T + p^{3} T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 - 34 T + p^{3} T^{2} )( 1 + 34 T + p^{3} T^{2} ) \) |
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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
−12.33646297102431111559339492183, −12.11917113986372416384165568040, −11.38027201171501602295893060526, −11.27092207244532764382203446368, −10.59045345612537717536911455391, −10.44328743866816546182325499652, −9.714500605771847873730943122607, −9.511890191610325524907150330140, −8.298635925901495190259452747950, −7.85455320404425822516531220183, −6.85669375606047361087021997131, −6.62108467337651406457482403153, −6.28073510557743450733848324910, −5.91692084694481231426959098201, −5.02300538229751352755386980884, −4.81451184439618695924021083826, −3.03480694906598043126778284700, −2.71111925951302792679487286244, −1.54143152584729375939984965322, −1.11050150136254401850123726940,
1.11050150136254401850123726940, 1.54143152584729375939984965322, 2.71111925951302792679487286244, 3.03480694906598043126778284700, 4.81451184439618695924021083826, 5.02300538229751352755386980884, 5.91692084694481231426959098201, 6.28073510557743450733848324910, 6.62108467337651406457482403153, 6.85669375606047361087021997131, 7.85455320404425822516531220183, 8.298635925901495190259452747950, 9.511890191610325524907150330140, 9.714500605771847873730943122607, 10.44328743866816546182325499652, 10.59045345612537717536911455391, 11.27092207244532764382203446368, 11.38027201171501602295893060526, 12.11917113986372416384165568040, 12.33646297102431111559339492183
