| L(s) = 1 | + 3-s + 1.84i·5-s + 9-s − 5.22i·11-s − 4.46i·13-s + 1.84i·15-s − 2.93i·17-s − 5.65·19-s − 2.16i·23-s + 1.58·25-s + 27-s − 5.41·29-s − 9.65·31-s − 5.22i·33-s + 1.41·37-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.826i·5-s + 0.333·9-s − 1.57i·11-s − 1.23i·13-s + 0.477i·15-s − 0.710i·17-s − 1.29·19-s − 0.451i·23-s + 0.317·25-s + 0.192·27-s − 1.00·29-s − 1.73·31-s − 0.909i·33-s + 0.232·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2352 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.156 + 0.987i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2352 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.156 + 0.987i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.428445176\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.428445176\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 7 | \( 1 \) |
| good | 5 | \( 1 - 1.84iT - 5T^{2} \) |
| 11 | \( 1 + 5.22iT - 11T^{2} \) |
| 13 | \( 1 + 4.46iT - 13T^{2} \) |
| 17 | \( 1 + 2.93iT - 17T^{2} \) |
| 19 | \( 1 + 5.65T + 19T^{2} \) |
| 23 | \( 1 + 2.16iT - 23T^{2} \) |
| 29 | \( 1 + 5.41T + 29T^{2} \) |
| 31 | \( 1 + 9.65T + 31T^{2} \) |
| 37 | \( 1 - 1.41T + 37T^{2} \) |
| 41 | \( 1 - 4.01iT - 41T^{2} \) |
| 43 | \( 1 + 3.06iT - 43T^{2} \) |
| 47 | \( 1 + 1.65T + 47T^{2} \) |
| 53 | \( 1 + 9.65T + 53T^{2} \) |
| 59 | \( 1 + 5.65T + 59T^{2} \) |
| 61 | \( 1 - 1.39iT - 61T^{2} \) |
| 67 | \( 1 + 13.5iT - 67T^{2} \) |
| 71 | \( 1 - 6.49iT - 71T^{2} \) |
| 73 | \( 1 + 4.90iT - 73T^{2} \) |
| 79 | \( 1 + 11.7iT - 79T^{2} \) |
| 83 | \( 1 - 17.6T + 83T^{2} \) |
| 89 | \( 1 - 9.05iT - 89T^{2} \) |
| 97 | \( 1 + 9.23iT - 97T^{2} \) |
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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
−8.750215095676581953554129469099, −8.000345614206699620818156837885, −7.37414978530115589804401672027, −6.40558395259215105392406140932, −5.79649754822801686159012134848, −4.76492584292363527288067393805, −3.44998011274917484814197161972, −3.15716297602895427291552467882, −2.07132934076936626575244825509, −0.41651503844867982351506706796,
1.66888168873993637368326272845, 2.10495074880529270675042426608, 3.69245251718891176062364560620, 4.36156762003876146327930896838, 5.00792745647270812800050564428, 6.16177882530729887930846486521, 7.03976915605220787966615662332, 7.64774276187023040115489618157, 8.591629891708448703535848966796, 9.180692932969937203752477910733
