| L(s) = 1 | + 0.898·2-s − 31.1·4-s + 25·5-s + 120.·7-s − 56.7·8-s + 22.4·10-s + 121·11-s + 667.·13-s + 108.·14-s + 947.·16-s + 74.5·17-s − 708.·19-s − 779.·20-s + 108.·22-s − 1.23e3·23-s + 625·25-s + 599.·26-s − 3.76e3·28-s + 3.85e3·29-s − 8.23e3·31-s + 2.66e3·32-s + 66.9·34-s + 3.01e3·35-s − 5.72e3·37-s − 636.·38-s − 1.41e3·40-s − 672.·41-s + ⋯ |
| L(s) = 1 | + 0.158·2-s − 0.974·4-s + 0.447·5-s + 0.931·7-s − 0.313·8-s + 0.0710·10-s + 0.301·11-s + 1.09·13-s + 0.147·14-s + 0.925·16-s + 0.0625·17-s − 0.450·19-s − 0.435·20-s + 0.0478·22-s − 0.488·23-s + 0.200·25-s + 0.173·26-s − 0.907·28-s + 0.851·29-s − 1.53·31-s + 0.460·32-s + 0.00993·34-s + 0.416·35-s − 0.687·37-s − 0.0714·38-s − 0.140·40-s − 0.0625·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 495 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 495 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(2.349617976\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.349617976\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 - 25T \) |
| 11 | \( 1 - 121T \) |
| good | 2 | \( 1 - 0.898T + 32T^{2} \) |
| 7 | \( 1 - 120.T + 1.68e4T^{2} \) |
| 13 | \( 1 - 667.T + 3.71e5T^{2} \) |
| 17 | \( 1 - 74.5T + 1.41e6T^{2} \) |
| 19 | \( 1 + 708.T + 2.47e6T^{2} \) |
| 23 | \( 1 + 1.23e3T + 6.43e6T^{2} \) |
| 29 | \( 1 - 3.85e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 8.23e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 5.72e3T + 6.93e7T^{2} \) |
| 41 | \( 1 + 672.T + 1.15e8T^{2} \) |
| 43 | \( 1 - 1.60e4T + 1.47e8T^{2} \) |
| 47 | \( 1 - 7.57e3T + 2.29e8T^{2} \) |
| 53 | \( 1 - 5.42e3T + 4.18e8T^{2} \) |
| 59 | \( 1 - 3.31e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 1.80e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 6.02e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 1.24e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 1.23e3T + 2.07e9T^{2} \) |
| 79 | \( 1 - 4.70e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 2.96e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 1.25e5T + 5.58e9T^{2} \) |
| 97 | \( 1 - 7.44e4T + 8.58e9T^{2} \) |
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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
−10.18204689557746357695884596975, −9.033540675237969093678854381987, −8.594292911294599714277025490544, −7.58505165738644771539166671290, −6.22340248994901766502810905740, −5.40442038898780070506974773580, −4.43450197763724417772823856842, −3.54072024489764627244085015654, −1.91087800068709205908355912498, −0.797944259346800784968155721282,
0.797944259346800784968155721282, 1.91087800068709205908355912498, 3.54072024489764627244085015654, 4.43450197763724417772823856842, 5.40442038898780070506974773580, 6.22340248994901766502810905740, 7.58505165738644771539166671290, 8.594292911294599714277025490544, 9.033540675237969093678854381987, 10.18204689557746357695884596975
