| L(s) = 1 | + 6i·3-s + 118i·7-s + 207·9-s − 192·11-s − 1.10e3i·13-s + 762i·17-s − 2.74e3·19-s − 708·21-s + 1.56e3i·23-s + 2.70e3i·27-s − 5.91e3·29-s + 6.86e3·31-s − 1.15e3i·33-s − 5.51e3i·37-s + 6.63e3·39-s + ⋯ |
| L(s) = 1 | + 0.384i·3-s + 0.910i·7-s + 0.851·9-s − 0.478·11-s − 1.81i·13-s + 0.639i·17-s − 1.74·19-s − 0.350·21-s + 0.617i·23-s + 0.712i·27-s − 1.30·29-s + 1.28·31-s − 0.184i·33-s − 0.662i·37-s + 0.698·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.5489922285\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5489922285\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| good | 3 | \( 1 - 6iT - 243T^{2} \) |
| 7 | \( 1 - 118iT - 1.68e4T^{2} \) |
| 11 | \( 1 + 192T + 1.61e5T^{2} \) |
| 13 | \( 1 + 1.10e3iT - 3.71e5T^{2} \) |
| 17 | \( 1 - 762iT - 1.41e6T^{2} \) |
| 19 | \( 1 + 2.74e3T + 2.47e6T^{2} \) |
| 23 | \( 1 - 1.56e3iT - 6.43e6T^{2} \) |
| 29 | \( 1 + 5.91e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 6.86e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 5.51e3iT - 6.93e7T^{2} \) |
| 41 | \( 1 + 378T + 1.15e8T^{2} \) |
| 43 | \( 1 + 2.43e3iT - 1.47e8T^{2} \) |
| 47 | \( 1 + 1.31e4iT - 2.29e8T^{2} \) |
| 53 | \( 1 - 9.17e3iT - 4.18e8T^{2} \) |
| 59 | \( 1 + 3.49e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 9.83e3T + 8.44e8T^{2} \) |
| 67 | \( 1 + 3.37e4iT - 1.35e9T^{2} \) |
| 71 | \( 1 + 7.02e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 2.19e4iT - 2.07e9T^{2} \) |
| 79 | \( 1 - 4.52e3T + 3.07e9T^{2} \) |
| 83 | \( 1 + 1.09e5iT - 3.93e9T^{2} \) |
| 89 | \( 1 + 3.84e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 1.91e3iT - 8.58e9T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.38470720424857789902980822801, −9.267046749788599409140316115193, −8.337072430850953816094280223988, −7.52558996808612879138538174576, −6.14385073006299033452150553282, −5.37586481750263093601978736692, −4.24000747607659299744162036485, −3.03819672808048996607263275868, −1.82081509963826339282955256905, −0.13227563984001086347377470682,
1.28351879081107145691186856313, 2.36449312996976240779456366108, 4.09944334565090916786839288226, 4.60986762400016188717843859286, 6.36007452042179233226218134565, 6.94627950645324395554567979354, 7.82660063868108157875934277220, 8.943295363554289286154383237678, 9.917125581878672098744037499220, 10.72241281150630916453178668730
