| L(s) = 1 | + 450·9-s − 384·11-s − 5.48e3·19-s − 1.18e4·29-s + 1.37e4·31-s − 756·41-s + 1.96e4·49-s − 6.99e4·59-s − 1.96e4·61-s − 1.40e5·71-s + 9.04e3·79-s + 1.43e5·81-s − 7.69e4·89-s − 1.72e5·99-s + 1.55e5·101-s − 4.13e5·109-s − 2.11e5·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 4.80e5·169-s + ⋯ |
| L(s) = 1 | + 1.85·9-s − 0.956·11-s − 3.48·19-s − 2.60·29-s + 2.56·31-s − 0.0702·41-s + 1.17·49-s − 2.61·59-s − 0.677·61-s − 3.30·71-s + 0.162·79-s + 2.42·81-s − 1.03·89-s − 1.77·99-s + 1.51·101-s − 3.33·109-s − 1.31·121-s + 5.50e−6·127-s + 5.09e−6·131-s + 4.55e−6·137-s + 4.38e−6·139-s + 3.69e−6·149-s + 3.56e−6·151-s + 3.23e−6·157-s + 2.94e−6·163-s + 2.77e−6·167-s − 1.29·169-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 160000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 160000 ^{s/2} \, \Gamma_{\C}(s+5/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.3013924670\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3013924670\) |
| \(L(\frac{7}{2})\) |
|
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 | | \( 1 \) |
| 5 | | \( 1 \) |
| good | 3 | $C_2^2$ | \( 1 - 50 p^{2} T^{2} + p^{10} T^{4} \) |
| 7 | $C_2^2$ | \( 1 - 19690 T^{2} + p^{10} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + 192 T + p^{5} T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 + 480650 T^{2} + p^{10} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 2259070 T^{2} + p^{10} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + 2740 T + p^{5} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 10420330 T^{2} + p^{10} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 5910 T + p^{5} T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 6868 T + p^{5} T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 108239590 T^{2} + p^{10} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 378 T + p^{5} T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 288092530 T^{2} + p^{10} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 286503130 T^{2} + p^{10} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 752228710 T^{2} + p^{10} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + 34980 T + p^{5} T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 9838 T + p^{5} T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 1563076930 T^{2} + p^{10} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 70212 T + p^{5} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 3662758990 T^{2} + p^{10} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 4520 T + p^{5} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 + 4019056190 T^{2} + p^{10} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 38490 T + p^{5} T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 17171001790 T^{2} + p^{10} T^{4} \) |
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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
−10.72241281150630916453178668730, −10.38470720424857789902980822801, −9.917125581878672098744037499220, −9.267046749788599409140316115193, −8.943295363554289286154383237678, −8.337072430850953816094280223988, −7.82660063868108157875934277220, −7.52558996808612879138538174576, −6.94627950645324395554567979354, −6.36007452042179233226218134565, −6.14385073006299033452150553282, −5.37586481750263093601978736692, −4.60986762400016188717843859286, −4.24000747607659299744162036485, −4.09944334565090916786839288226, −3.03819672808048996607263275868, −2.36449312996976240779456366108, −1.82081509963826339282955256905, −1.28351879081107145691186856313, −0.13227563984001086347377470682,
0.13227563984001086347377470682, 1.28351879081107145691186856313, 1.82081509963826339282955256905, 2.36449312996976240779456366108, 3.03819672808048996607263275868, 4.09944334565090916786839288226, 4.24000747607659299744162036485, 4.60986762400016188717843859286, 5.37586481750263093601978736692, 6.14385073006299033452150553282, 6.36007452042179233226218134565, 6.94627950645324395554567979354, 7.52558996808612879138538174576, 7.82660063868108157875934277220, 8.337072430850953816094280223988, 8.943295363554289286154383237678, 9.267046749788599409140316115193, 9.917125581878672098744037499220, 10.38470720424857789902980822801, 10.72241281150630916453178668730
