| L(s) = 1 | − 12·3-s + 54·5-s + 243·9-s − 540·11-s + 836·13-s − 648·15-s + 594·17-s + 836·19-s + 4.10e3·23-s + 3.12e3·25-s − 7.02e3·27-s − 1.18e3·29-s + 4.25e3·31-s + 6.48e3·33-s + 298·37-s − 1.00e4·39-s − 3.44e4·41-s − 2.42e4·43-s + 1.31e4·45-s − 1.29e3·47-s − 7.12e3·51-s − 1.94e4·53-s − 2.91e4·55-s − 1.00e4·57-s − 7.66e3·59-s − 3.47e4·61-s + 4.51e4·65-s + ⋯ |
| L(s) = 1 | − 0.769·3-s + 0.965·5-s + 9-s − 1.34·11-s + 1.37·13-s − 0.743·15-s + 0.498·17-s + 0.531·19-s + 1.61·23-s + 25-s − 1.85·27-s − 0.262·29-s + 0.795·31-s + 1.03·33-s + 0.0357·37-s − 1.05·39-s − 3.20·41-s − 1.99·43-s + 0.965·45-s − 0.0855·47-s − 0.383·51-s − 0.953·53-s − 1.29·55-s − 0.408·57-s − 0.286·59-s − 1.19·61-s + 1.32·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 38416 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 38416 ^{s/2} \, \Gamma_{\C}(s+5/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(2.188032326\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.188032326\) |
| \(L(\frac{7}{2})\) |
|
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 | | \( 1 \) |
| 7 | | \( 1 \) |
| good | 3 | $C_2^2$ | \( 1 + 4 p T - 11 p^{2} T^{2} + 4 p^{6} T^{3} + p^{10} T^{4} \) |
| 5 | $C_2^2$ | \( 1 - 54 T - 209 T^{2} - 54 p^{5} T^{3} + p^{10} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 540 T + 130549 T^{2} + 540 p^{5} T^{3} + p^{10} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 418 T + p^{5} T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 - 594 T - 1067021 T^{2} - 594 p^{5} T^{3} + p^{10} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 44 p T - 4923 p^{2} T^{2} - 44 p^{6} T^{3} + p^{10} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 4104 T + 10406473 T^{2} - 4104 p^{5} T^{3} + p^{10} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 594 T + p^{5} T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 - 4256 T - 10515615 T^{2} - 4256 p^{5} T^{3} + p^{10} T^{4} \) |
| 37 | $C_2^2$ | \( 1 - 298 T - 69255153 T^{2} - 298 p^{5} T^{3} + p^{10} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 17226 T + p^{5} T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + 12100 T + p^{5} T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 + 1296 T - 227665391 T^{2} + 1296 p^{5} T^{3} + p^{10} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 19494 T - 38179457 T^{2} + 19494 p^{5} T^{3} + p^{10} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 7668 T - 656126075 T^{2} + 7668 p^{5} T^{3} + p^{10} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 34738 T + 362132343 T^{2} + 34738 p^{5} T^{3} + p^{10} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 21812 T - 874361763 T^{2} + 21812 p^{5} T^{3} + p^{10} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 46872 T + p^{5} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 67562 T + 2491552251 T^{2} - 67562 p^{5} T^{3} + p^{10} T^{4} \) |
| 79 | $C_2^2$ | \( 1 - 76912 T + 2838399345 T^{2} - 76912 p^{5} T^{3} + p^{10} T^{4} \) |
| 83 | $C_2$ | \( ( 1 + 67716 T + p^{5} T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - 29754 T - 4698758933 T^{2} - 29754 p^{5} T^{3} + p^{10} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 122398 T + p^{5} T^{2} )^{2} \) |
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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
−11.85119107672026696646535185627, −11.26209083439098568390253517226, −10.87882798990519051768630790944, −10.26348006073767225104631589088, −10.14860774335752357848964180539, −9.503386557880415555607349487656, −8.950271390804328155913123521383, −8.279499185353390168756216445291, −7.87666714272711221776639454973, −6.92526924955320658115343795397, −6.84012529087407548993483271901, −5.93142795273266138445825588925, −5.64245290954351691803255546160, −4.92465214809842891543103173085, −4.68977070612841135500151565035, −3.24102611726558864429364232170, −3.24064691863337616706659543766, −1.79740224768445110230891669265, −1.45734951064192139503349566474, −0.49213098530591072928089558796,
0.49213098530591072928089558796, 1.45734951064192139503349566474, 1.79740224768445110230891669265, 3.24064691863337616706659543766, 3.24102611726558864429364232170, 4.68977070612841135500151565035, 4.92465214809842891543103173085, 5.64245290954351691803255546160, 5.93142795273266138445825588925, 6.84012529087407548993483271901, 6.92526924955320658115343795397, 7.87666714272711221776639454973, 8.279499185353390168756216445291, 8.950271390804328155913123521383, 9.503386557880415555607349487656, 10.14860774335752357848964180539, 10.26348006073767225104631589088, 10.87882798990519051768630790944, 11.26209083439098568390253517226, 11.85119107672026696646535185627
