| L(s) = 1 | − 2-s + 4-s − 5-s − 1.58i·7-s − 8-s + 10-s − 0.711·11-s + 6.95i·13-s + 1.58i·14-s + 16-s + 5.31i·17-s − 1.17·19-s − 20-s + 0.711·22-s + 5.40i·23-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.5·4-s − 0.447·5-s − 0.600i·7-s − 0.353·8-s + 0.316·10-s − 0.214·11-s + 1.93i·13-s + 0.424i·14-s + 0.250·16-s + 1.28i·17-s − 0.269·19-s − 0.223·20-s + 0.151·22-s + 1.12i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6030 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.921 + 0.387i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6030 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.921 + 0.387i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.1236117559\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1236117559\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| 67 | \( 1 + (-1.76 + 7.99i)T \) |
| good | 7 | \( 1 + 1.58iT - 7T^{2} \) |
| 11 | \( 1 + 0.711T + 11T^{2} \) |
| 13 | \( 1 - 6.95iT - 13T^{2} \) |
| 17 | \( 1 - 5.31iT - 17T^{2} \) |
| 19 | \( 1 + 1.17T + 19T^{2} \) |
| 23 | \( 1 - 5.40iT - 23T^{2} \) |
| 29 | \( 1 - 4.19iT - 29T^{2} \) |
| 31 | \( 1 + 5.83iT - 31T^{2} \) |
| 37 | \( 1 + 5.35T + 37T^{2} \) |
| 41 | \( 1 + 6.36T + 41T^{2} \) |
| 43 | \( 1 + 7.11iT - 43T^{2} \) |
| 47 | \( 1 - 13.0iT - 47T^{2} \) |
| 53 | \( 1 + 3.99T + 53T^{2} \) |
| 59 | \( 1 + 12.3iT - 59T^{2} \) |
| 61 | \( 1 - 11.4iT - 61T^{2} \) |
| 71 | \( 1 - 4.17iT - 71T^{2} \) |
| 73 | \( 1 - 14.5T + 73T^{2} \) |
| 79 | \( 1 + 8.91iT - 79T^{2} \) |
| 83 | \( 1 + 16.7iT - 83T^{2} \) |
| 89 | \( 1 + 12.9iT - 89T^{2} \) |
| 97 | \( 1 + 0.688iT - 97T^{2} \) |
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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
−8.489591433789607588555403357586, −7.77001335009474412883963459776, −7.14344644256637806232944380615, −6.58685428572620112788833021218, −5.82897595753994761713456898519, −4.74412411983570570736397371771, −3.99340867836909910176782431775, −3.39075416268328108951455591184, −2.03506182308823033164888833190, −1.44293551565673610161122314454,
0.04707557193874247927906530881, 0.952803914006507287154109076518, 2.41246309600012503236844365753, 2.90533084684247772905738677330, 3.81544371569824971910223261250, 5.15073753136313003139622071682, 5.32699983144492687688786220588, 6.49531343361409253836641329415, 7.01214818927925809588312004110, 8.023010611666337068754120874783
