| 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
| \(L(2)\) |
\(\approx\) |
\(5.278109538\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.278109538\) |
| \(L(\frac{5}{2})\) |
|
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
−12.36641398324479736710698441264, −12.02095942525041426445196955830, −11.55826329382467025742075604558, −11.49795430180184384919655960328, −10.49727631208475665347788053946, −10.12554484334044331767555938503, −9.838347046414425363404974006436, −8.652957594784315875732671652393, −8.258781037017152740684693287145, −8.020996584931673960265912696484, −7.52786014205670968471347668629, −7.01981618757721621488745375285, −6.45808700105223890889163867485, −5.87316744889412374597264916811, −4.61832234584018278754661254698, −3.91863330079337806323822772989, −3.36893076057912630994637160177, −2.67798950454336455685485603753, −2.20416761408218747990590335667, −1.01935487348965279122303313242,
1.01935487348965279122303313242, 2.20416761408218747990590335667, 2.67798950454336455685485603753, 3.36893076057912630994637160177, 3.91863330079337806323822772989, 4.61832234584018278754661254698, 5.87316744889412374597264916811, 6.45808700105223890889163867485, 7.01981618757721621488745375285, 7.52786014205670968471347668629, 8.020996584931673960265912696484, 8.258781037017152740684693287145, 8.652957594784315875732671652393, 9.838347046414425363404974006436, 10.12554484334044331767555938503, 10.49727631208475665347788053946, 11.49795430180184384919655960328, 11.55826329382467025742075604558, 12.02095942525041426445196955830, 12.36641398324479736710698441264
