| L(s) = 1 | + 1.73·5-s + (1 − 2.44i)7-s − 4.24i·11-s − 2.44i·13-s + 1.73·17-s − 2.44i·19-s − 8.48i·23-s − 2.00·25-s + 4.24i·29-s + 7.34i·31-s + (1.73 − 4.24i)35-s + 37-s − 1.73·41-s − 7·43-s − 12.1·47-s + ⋯ |
| L(s) = 1 | + 0.774·5-s + (0.377 − 0.925i)7-s − 1.27i·11-s − 0.679i·13-s + 0.420·17-s − 0.561i·19-s − 1.76i·23-s − 0.400·25-s + 0.787i·29-s + 1.31i·31-s + (0.292 − 0.717i)35-s + 0.164·37-s − 0.270·41-s − 1.06·43-s − 1.76·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.377 + 0.925i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.377 + 0.925i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.860987433\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.860987433\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (-1 + 2.44i)T \) |
| good | 5 | \( 1 - 1.73T + 5T^{2} \) |
| 11 | \( 1 + 4.24iT - 11T^{2} \) |
| 13 | \( 1 + 2.44iT - 13T^{2} \) |
| 17 | \( 1 - 1.73T + 17T^{2} \) |
| 19 | \( 1 + 2.44iT - 19T^{2} \) |
| 23 | \( 1 + 8.48iT - 23T^{2} \) |
| 29 | \( 1 - 4.24iT - 29T^{2} \) |
| 31 | \( 1 - 7.34iT - 31T^{2} \) |
| 37 | \( 1 - T + 37T^{2} \) |
| 41 | \( 1 + 1.73T + 41T^{2} \) |
| 43 | \( 1 + 7T + 43T^{2} \) |
| 47 | \( 1 + 12.1T + 47T^{2} \) |
| 53 | \( 1 - 53T^{2} \) |
| 59 | \( 1 + 8.66T + 59T^{2} \) |
| 61 | \( 1 - 2.44iT - 61T^{2} \) |
| 67 | \( 1 - 10T + 67T^{2} \) |
| 71 | \( 1 + 8.48iT - 71T^{2} \) |
| 73 | \( 1 - 9.79iT - 73T^{2} \) |
| 79 | \( 1 + 5T + 79T^{2} \) |
| 83 | \( 1 - 12.1T + 83T^{2} \) |
| 89 | \( 1 - 10.3T + 89T^{2} \) |
| 97 | \( 1 + 2.44iT - 97T^{2} \) |
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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
−8.393429511180140727998793496278, −7.924585570759252383672264222530, −6.79578460914676005506353531107, −6.34299734581220373938676842747, −5.33164673248896392677464896767, −4.77448678283064511973801215807, −3.57869965554652195770290364262, −2.88737819931145486457547640673, −1.59709802105164168344297905258, −0.55503927228159515059183686625,
1.75770427776262166650486565344, 2.01208597523022734434091820916, 3.32432077125894175018974792028, 4.39680392659725505441431372008, 5.21012889090947761970081452266, 5.87058944076916705566913397397, 6.57124236584979775426725021075, 7.59772322587017670794869649115, 8.093035265933463506337022431160, 9.179670298996030349200139633682
