| L(s) = 1 | − 0.206·5-s − 3.03i·7-s − 5.02i·11-s − 4.03i·13-s − 4.81i·17-s + 1.03·19-s + 5.43·23-s − 4.95·25-s − 2.90·29-s + 1.23i·31-s + 0.629i·35-s − 8.77i·37-s + 0.979i·41-s − 3.18·43-s − 3.26·47-s + ⋯ |
| L(s) = 1 | − 0.0925·5-s − 1.14i·7-s − 1.51i·11-s − 1.12i·13-s − 1.16i·17-s + 0.238·19-s + 1.13·23-s − 0.991·25-s − 0.539·29-s + 0.222i·31-s + 0.106i·35-s − 1.44i·37-s + 0.152i·41-s − 0.485·43-s − 0.476·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5184 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.965 + 0.258i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5184 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.965 + 0.258i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.382531371\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.382531371\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 + 0.206T + 5T^{2} \) |
| 7 | \( 1 + 3.03iT - 7T^{2} \) |
| 11 | \( 1 + 5.02iT - 11T^{2} \) |
| 13 | \( 1 + 4.03iT - 13T^{2} \) |
| 17 | \( 1 + 4.81iT - 17T^{2} \) |
| 19 | \( 1 - 1.03T + 19T^{2} \) |
| 23 | \( 1 - 5.43T + 23T^{2} \) |
| 29 | \( 1 + 2.90T + 29T^{2} \) |
| 31 | \( 1 - 1.23iT - 31T^{2} \) |
| 37 | \( 1 + 8.77iT - 37T^{2} \) |
| 41 | \( 1 - 0.979iT - 41T^{2} \) |
| 43 | \( 1 + 3.18T + 43T^{2} \) |
| 47 | \( 1 + 3.26T + 47T^{2} \) |
| 53 | \( 1 - 7.91T + 53T^{2} \) |
| 59 | \( 1 - 14.8iT - 59T^{2} \) |
| 61 | \( 1 + 9.53iT - 61T^{2} \) |
| 67 | \( 1 + 0.735T + 67T^{2} \) |
| 71 | \( 1 - 2.33T + 71T^{2} \) |
| 73 | \( 1 + 10.0T + 73T^{2} \) |
| 79 | \( 1 - 14.1iT - 79T^{2} \) |
| 83 | \( 1 + 4.23iT - 83T^{2} \) |
| 89 | \( 1 - 11.6iT - 89T^{2} \) |
| 97 | \( 1 - 4.84T + 97T^{2} \) |
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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
−7.62884241703692328179904900051, −7.43234283651169703808531359716, −6.47459102861714637178814943576, −5.60657575513943269635654583570, −5.09941689248961088541043860683, −3.97410291685198343976328441060, −3.39432776716368442697553650449, −2.62252280790773332562563477872, −1.06807910886186265715116436215, −0.40459247275964784248524072429,
1.65583881657798331230057147403, 2.12223622439714095164209556152, 3.22711129069632908826728474463, 4.21351895260500669675166760156, 4.83652708836835293672172347675, 5.65603267339175030703536712797, 6.42202359951755023817240125752, 7.07210420520354446166349657989, 7.81228073003958451232164507469, 8.651659419646096172710807289079
