| L(s) = 1 | − 3.23i·5-s − 4.57i·7-s − 1.23i·11-s + (2.23 + 2.82i)13-s − 5.65·17-s − 1.08i·19-s + 3.49·23-s − 5.47·25-s − 9.15·29-s + 4.57i·31-s − 14.8·35-s + 5.65i·37-s − 5.70i·41-s + 8.94·43-s + 2.76i·47-s + ⋯ |
| L(s) = 1 | − 1.44i·5-s − 1.72i·7-s − 0.372i·11-s + (0.620 + 0.784i)13-s − 1.37·17-s − 0.247i·19-s + 0.728·23-s − 1.09·25-s − 1.69·29-s + 0.821i·31-s − 2.50·35-s + 0.929i·37-s − 0.891i·41-s + 1.36·43-s + 0.403i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 936 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.784 + 0.620i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 936 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.784 + 0.620i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.414175 - 1.19173i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.414175 - 1.19173i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 13 | \( 1 + (-2.23 - 2.82i)T \) |
| good | 5 | \( 1 + 3.23iT - 5T^{2} \) |
| 7 | \( 1 + 4.57iT - 7T^{2} \) |
| 11 | \( 1 + 1.23iT - 11T^{2} \) |
| 17 | \( 1 + 5.65T + 17T^{2} \) |
| 19 | \( 1 + 1.08iT - 19T^{2} \) |
| 23 | \( 1 - 3.49T + 23T^{2} \) |
| 29 | \( 1 + 9.15T + 29T^{2} \) |
| 31 | \( 1 - 4.57iT - 31T^{2} \) |
| 37 | \( 1 - 5.65iT - 37T^{2} \) |
| 41 | \( 1 + 5.70iT - 41T^{2} \) |
| 43 | \( 1 - 8.94T + 43T^{2} \) |
| 47 | \( 1 - 2.76iT - 47T^{2} \) |
| 53 | \( 1 - 3.49T + 53T^{2} \) |
| 59 | \( 1 + 11.7iT - 59T^{2} \) |
| 61 | \( 1 + 0.472T + 61T^{2} \) |
| 67 | \( 1 + 10.2iT - 67T^{2} \) |
| 71 | \( 1 + 7.70iT - 71T^{2} \) |
| 73 | \( 1 - 73T^{2} \) |
| 79 | \( 1 - 16.9T + 79T^{2} \) |
| 83 | \( 1 + 6.76iT - 83T^{2} \) |
| 89 | \( 1 - 15.2iT - 89T^{2} \) |
| 97 | \( 1 + 6.99iT - 97T^{2} \) |
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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
−9.463882339699247855379806420453, −9.008600239249340245377003526900, −8.132374588141535124806760948853, −7.19000027634811692116507058201, −6.40867365754002606513564113671, −5.09657142650075933718599508208, −4.37231684859331620534169954490, −3.64740056402293089991722686318, −1.68923489789662812030824490776, −0.59089290338068428532584710835,
2.19162620061660484556994145304, 2.80196077961847905531398666849, 3.96971296506311400058308649747, 5.46201196152039593459758362742, 6.04193141510908078809741373484, 6.93065338843268042146262681252, 7.81863948535289625820677115073, 8.862165805946161208810200141134, 9.433568733929300631426892102153, 10.56358952367817386444860941209
