| L(s) = 1 | − 4.04i·5-s + 3.59·7-s + 4.29i·11-s + 1.77i·13-s − 2.86·17-s + i·19-s + 2.74·23-s − 11.3·25-s − 2.29i·29-s − 2.33·31-s − 14.5i·35-s − 8.06i·37-s + 2.85·41-s − 0.241i·43-s − 0.191·47-s + ⋯ |
| L(s) = 1 | − 1.81i·5-s + 1.35·7-s + 1.29i·11-s + 0.491i·13-s − 0.695·17-s + 0.229i·19-s + 0.571·23-s − 2.27·25-s − 0.425i·29-s − 0.418·31-s − 2.45i·35-s − 1.32i·37-s + 0.446·41-s − 0.0368i·43-s − 0.0280·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5472 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.129 + 0.991i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5472 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.129 + 0.991i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.030782323\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.030782323\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 19 | \( 1 - iT \) |
| good | 5 | \( 1 + 4.04iT - 5T^{2} \) |
| 7 | \( 1 - 3.59T + 7T^{2} \) |
| 11 | \( 1 - 4.29iT - 11T^{2} \) |
| 13 | \( 1 - 1.77iT - 13T^{2} \) |
| 17 | \( 1 + 2.86T + 17T^{2} \) |
| 23 | \( 1 - 2.74T + 23T^{2} \) |
| 29 | \( 1 + 2.29iT - 29T^{2} \) |
| 31 | \( 1 + 2.33T + 31T^{2} \) |
| 37 | \( 1 + 8.06iT - 37T^{2} \) |
| 41 | \( 1 - 2.85T + 41T^{2} \) |
| 43 | \( 1 + 0.241iT - 43T^{2} \) |
| 47 | \( 1 + 0.191T + 47T^{2} \) |
| 53 | \( 1 + 8.10iT - 53T^{2} \) |
| 59 | \( 1 + 4.36iT - 59T^{2} \) |
| 61 | \( 1 + 15.3iT - 61T^{2} \) |
| 67 | \( 1 + 13.2iT - 67T^{2} \) |
| 71 | \( 1 - 7.44T + 71T^{2} \) |
| 73 | \( 1 - 9.20T + 73T^{2} \) |
| 79 | \( 1 + 8.96T + 79T^{2} \) |
| 83 | \( 1 + 1.94iT - 83T^{2} \) |
| 89 | \( 1 - 3.24T + 89T^{2} \) |
| 97 | \( 1 - 17.6T + 97T^{2} \) |
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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
−8.027654021919584970840838014099, −7.50006446346856813364478492383, −6.57242179422290541836868973993, −5.48994158402561199997534824293, −4.89380771950856799300113596802, −4.55109407548817997238758237710, −3.82564497241690455761309870314, −1.94960356467714905929678777549, −1.84224202543442161418617288640, −0.56009307114104398372159428709,
1.14040269592334438895752644825, 2.37438528701258058263745282096, 2.96544147627196691174192349658, 3.75804075278148107066832579292, 4.70967669215794353949868522832, 5.62526630171408614147288711140, 6.20704921826268267403995175953, 7.04490435356467415747358789800, 7.52265680504617794348745911838, 8.330825275168325029788482160060
