| L(s) = 1 | + 2.68·3-s + 2.68·5-s + 7-s + 4.18·9-s + 11-s − 2.50·13-s + 7.18·15-s − 6.37·17-s − 3.87·19-s + 2.68·21-s − 4.55·23-s + 2.18·25-s + 3.18·27-s + 3.01·29-s + 5.18·31-s + 2.68·33-s + 2.68·35-s − 6.55·37-s − 6.72·39-s − 4.34·41-s − 1.01·43-s + 11.2·45-s + 0.637·47-s + 49-s − 17.1·51-s − 3.01·53-s + 2.68·55-s + ⋯ |
| L(s) = 1 | + 1.54·3-s + 1.19·5-s + 0.377·7-s + 1.39·9-s + 0.301·11-s − 0.695·13-s + 1.85·15-s − 1.54·17-s − 0.888·19-s + 0.585·21-s − 0.949·23-s + 0.437·25-s + 0.613·27-s + 0.560·29-s + 0.932·31-s + 0.466·33-s + 0.453·35-s − 1.07·37-s − 1.07·39-s − 0.678·41-s − 0.155·43-s + 1.67·45-s + 0.0929·47-s + 0.142·49-s − 2.39·51-s − 0.414·53-s + 0.361·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 616 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 616 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.841356521\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.841356521\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 - T \) |
| good | 3 | \( 1 - 2.68T + 3T^{2} \) |
| 5 | \( 1 - 2.68T + 5T^{2} \) |
| 13 | \( 1 + 2.50T + 13T^{2} \) |
| 17 | \( 1 + 6.37T + 17T^{2} \) |
| 19 | \( 1 + 3.87T + 19T^{2} \) |
| 23 | \( 1 + 4.55T + 23T^{2} \) |
| 29 | \( 1 - 3.01T + 29T^{2} \) |
| 31 | \( 1 - 5.18T + 31T^{2} \) |
| 37 | \( 1 + 6.55T + 37T^{2} \) |
| 41 | \( 1 + 4.34T + 41T^{2} \) |
| 43 | \( 1 + 1.01T + 43T^{2} \) |
| 47 | \( 1 - 0.637T + 47T^{2} \) |
| 53 | \( 1 + 3.01T + 53T^{2} \) |
| 59 | \( 1 - 12.0T + 59T^{2} \) |
| 61 | \( 1 - 15.6T + 61T^{2} \) |
| 67 | \( 1 - 5.56T + 67T^{2} \) |
| 71 | \( 1 - 11.5T + 71T^{2} \) |
| 73 | \( 1 + 13.3T + 73T^{2} \) |
| 79 | \( 1 - 6.37T + 79T^{2} \) |
| 83 | \( 1 - 2.50T + 83T^{2} \) |
| 89 | \( 1 + 12.9T + 89T^{2} \) |
| 97 | \( 1 + 10.8T + 97T^{2} \) |
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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
−10.17200699564353002570535707285, −9.767410128491079068465780218256, −8.698592453568943557245085899711, −8.404078271664583362594909700066, −7.11304303166610787336314853109, −6.28821758971901757156237277195, −4.92633633218914971899591074776, −3.88883278231138769830239128868, −2.43611974768842675042505582867, −1.96478652796702649110347377046,
1.96478652796702649110347377046, 2.43611974768842675042505582867, 3.88883278231138769830239128868, 4.92633633218914971899591074776, 6.28821758971901757156237277195, 7.11304303166610787336314853109, 8.404078271664583362594909700066, 8.698592453568943557245085899711, 9.767410128491079068465780218256, 10.17200699564353002570535707285
