| L(s) = 1 | + 3.04·5-s + 0.521·11-s − 0.737·17-s + 4.41·19-s + 4.82·23-s + 4.24·25-s + 8.59·29-s + 3·31-s − 3.24·37-s − 9.85·41-s − 8.24·43-s + 6.81·47-s − 1.04·53-s + 1.58·55-s − 11.4·59-s + 0.343·61-s + 8.48·67-s + 9.12·71-s + 4.58·73-s − 8.24·79-s + 6.08·83-s − 2.24·85-s + 9.85·89-s + 13.4·95-s + 5.65·97-s + 12.8·101-s + 17.8·103-s + ⋯ |
| L(s) = 1 | + 1.35·5-s + 0.157·11-s − 0.178·17-s + 1.01·19-s + 1.00·23-s + 0.848·25-s + 1.59·29-s + 0.538·31-s − 0.533·37-s − 1.53·41-s − 1.25·43-s + 0.994·47-s − 0.143·53-s + 0.213·55-s − 1.48·59-s + 0.0439·61-s + 1.03·67-s + 1.08·71-s + 0.536·73-s − 0.927·79-s + 0.667·83-s − 0.243·85-s + 1.04·89-s + 1.37·95-s + 0.574·97-s + 1.28·101-s + 1.75·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5292 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5292 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.863312494\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.863312494\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| good | 5 | \( 1 - 3.04T + 5T^{2} \) |
| 11 | \( 1 - 0.521T + 11T^{2} \) |
| 13 | \( 1 + 13T^{2} \) |
| 17 | \( 1 + 0.737T + 17T^{2} \) |
| 19 | \( 1 - 4.41T + 19T^{2} \) |
| 23 | \( 1 - 4.82T + 23T^{2} \) |
| 29 | \( 1 - 8.59T + 29T^{2} \) |
| 31 | \( 1 - 3T + 31T^{2} \) |
| 37 | \( 1 + 3.24T + 37T^{2} \) |
| 41 | \( 1 + 9.85T + 41T^{2} \) |
| 43 | \( 1 + 8.24T + 43T^{2} \) |
| 47 | \( 1 - 6.81T + 47T^{2} \) |
| 53 | \( 1 + 1.04T + 53T^{2} \) |
| 59 | \( 1 + 11.4T + 59T^{2} \) |
| 61 | \( 1 - 0.343T + 61T^{2} \) |
| 67 | \( 1 - 8.48T + 67T^{2} \) |
| 71 | \( 1 - 9.12T + 71T^{2} \) |
| 73 | \( 1 - 4.58T + 73T^{2} \) |
| 79 | \( 1 + 8.24T + 79T^{2} \) |
| 83 | \( 1 - 6.08T + 83T^{2} \) |
| 89 | \( 1 - 9.85T + 89T^{2} \) |
| 97 | \( 1 - 5.65T + 97T^{2} \) |
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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
−8.332338822062878778160260222885, −7.36873468040315950319870731974, −6.59235175297084773416132831746, −6.17124026456080164968087891756, −5.12132052324490552952935861456, −4.89782935034720669000825193702, −3.53137700869910518201416047705, −2.78533436980712813963835495937, −1.86258384872020472942099343449, −0.965812973016359720873536583886,
0.965812973016359720873536583886, 1.86258384872020472942099343449, 2.78533436980712813963835495937, 3.53137700869910518201416047705, 4.89782935034720669000825193702, 5.12132052324490552952935861456, 6.17124026456080164968087891756, 6.59235175297084773416132831746, 7.36873468040315950319870731974, 8.332338822062878778160260222885
