| L(s) = 1 | − 2.73·3-s + 5-s + 4.46·9-s − 2·11-s + 0.732·13-s − 2.73·15-s + 7.46·17-s − 19-s − 1.46·23-s + 25-s − 3.99·27-s + 3.46·29-s + 4·31-s + 5.46·33-s − 7.66·37-s − 2·39-s + 0.535·41-s + 2.92·43-s + 4.46·45-s − 2.92·47-s − 7·49-s − 20.3·51-s + 11.6·53-s − 2·55-s + 2.73·57-s − 1.46·59-s + 6.53·61-s + ⋯ |
| L(s) = 1 | − 1.57·3-s + 0.447·5-s + 1.48·9-s − 0.603·11-s + 0.203·13-s − 0.705·15-s + 1.81·17-s − 0.229·19-s − 0.305·23-s + 0.200·25-s − 0.769·27-s + 0.643·29-s + 0.718·31-s + 0.951·33-s − 1.25·37-s − 0.320·39-s + 0.0836·41-s + 0.446·43-s + 0.665·45-s − 0.427·47-s − 49-s − 2.85·51-s + 1.60·53-s − 0.269·55-s + 0.361·57-s − 0.190·59-s + 0.836·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6080 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6080 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.140044551\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.140044551\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 - T \) |
| 19 | \( 1 + T \) |
| good | 3 | \( 1 + 2.73T + 3T^{2} \) |
| 7 | \( 1 + 7T^{2} \) |
| 11 | \( 1 + 2T + 11T^{2} \) |
| 13 | \( 1 - 0.732T + 13T^{2} \) |
| 17 | \( 1 - 7.46T + 17T^{2} \) |
| 23 | \( 1 + 1.46T + 23T^{2} \) |
| 29 | \( 1 - 3.46T + 29T^{2} \) |
| 31 | \( 1 - 4T + 31T^{2} \) |
| 37 | \( 1 + 7.66T + 37T^{2} \) |
| 41 | \( 1 - 0.535T + 41T^{2} \) |
| 43 | \( 1 - 2.92T + 43T^{2} \) |
| 47 | \( 1 + 2.92T + 47T^{2} \) |
| 53 | \( 1 - 11.6T + 53T^{2} \) |
| 59 | \( 1 + 1.46T + 59T^{2} \) |
| 61 | \( 1 - 6.53T + 61T^{2} \) |
| 67 | \( 1 - 4.19T + 67T^{2} \) |
| 71 | \( 1 - 10.9T + 71T^{2} \) |
| 73 | \( 1 + 10.3T + 73T^{2} \) |
| 79 | \( 1 + 8.39T + 79T^{2} \) |
| 83 | \( 1 - 5.46T + 83T^{2} \) |
| 89 | \( 1 + 3.46T + 89T^{2} \) |
| 97 | \( 1 + 15.6T + 97T^{2} \) |
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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
−7.989016757283442827782244495505, −7.15984503953733026199042122562, −6.50864472877248975966058036889, −5.79864842048174225544277567356, −5.36980063659905628613049227738, −4.76136496544171180091690301047, −3.76487143304069908929213538718, −2.75930834882753997711101508291, −1.53257382457574985707899959633, −0.64397199377802081986455331939,
0.64397199377802081986455331939, 1.53257382457574985707899959633, 2.75930834882753997711101508291, 3.76487143304069908929213538718, 4.76136496544171180091690301047, 5.36980063659905628613049227738, 5.79864842048174225544277567356, 6.50864472877248975966058036889, 7.15984503953733026199042122562, 7.989016757283442827782244495505
