| L(s) = 1 | − 2·2-s + 4·4-s − 8.24·5-s + 8.24·7-s − 8·8-s + 9·9-s + 16.4·10-s − 16.4·14-s + 16·16-s + 17·17-s − 18·18-s + 30·19-s − 32.9·20-s + 8.24·23-s + 43·25-s + 32.9·28-s + 57.7·29-s − 57.7·31-s − 32·32-s − 34·34-s − 68·35-s + 36·36-s − 8.24·37-s − 60·38-s + 65.9·40-s − 50·43-s − 74.2·45-s + ⋯ |
| L(s) = 1 | − 2-s + 4-s − 1.64·5-s + 1.17·7-s − 8-s + 9-s + 1.64·10-s − 1.17·14-s + 16-s + 17-s − 18-s + 1.57·19-s − 1.64·20-s + 0.358·23-s + 1.71·25-s + 1.17·28-s + 1.99·29-s − 1.86·31-s − 32-s − 34-s − 1.94·35-s + 36-s − 0.222·37-s − 1.57·38-s + 1.64·40-s − 1.16·43-s − 1.64·45-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 136 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 136 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.8681340669\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8681340669\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 2T \) |
| 17 | \( 1 - 17T \) |
| good | 3 | \( 1 - 9T^{2} \) |
| 5 | \( 1 + 8.24T + 25T^{2} \) |
| 7 | \( 1 - 8.24T + 49T^{2} \) |
| 11 | \( 1 - 121T^{2} \) |
| 13 | \( 1 - 169T^{2} \) |
| 19 | \( 1 - 30T + 361T^{2} \) |
| 23 | \( 1 - 8.24T + 529T^{2} \) |
| 29 | \( 1 - 57.7T + 841T^{2} \) |
| 31 | \( 1 + 57.7T + 961T^{2} \) |
| 37 | \( 1 + 8.24T + 1.36e3T^{2} \) |
| 41 | \( 1 - 1.68e3T^{2} \) |
| 43 | \( 1 + 50T + 1.84e3T^{2} \) |
| 47 | \( 1 - 2.20e3T^{2} \) |
| 53 | \( 1 - 2.80e3T^{2} \) |
| 59 | \( 1 + 18T + 3.48e3T^{2} \) |
| 61 | \( 1 - 57.7T + 3.72e3T^{2} \) |
| 67 | \( 1 + 66T + 4.48e3T^{2} \) |
| 71 | \( 1 - 140.T + 5.04e3T^{2} \) |
| 73 | \( 1 - 5.32e3T^{2} \) |
| 79 | \( 1 + 57.7T + 6.24e3T^{2} \) |
| 83 | \( 1 - 30T + 6.88e3T^{2} \) |
| 89 | \( 1 + 110T + 7.92e3T^{2} \) |
| 97 | \( 1 - 9.40e3T^{2} \) |
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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
−12.44208745103299968706643742150, −11.75928771755263924966141257689, −11.00481433245061999685969649494, −9.902078465133407039524283585983, −8.510613124928685313819846009109, −7.70708992122614514332932794804, −7.11425834187545914089965049765, −5.00745163781477637100818662976, −3.47597569964943515944581694148, −1.16606438928378329232048021513,
1.16606438928378329232048021513, 3.47597569964943515944581694148, 5.00745163781477637100818662976, 7.11425834187545914089965049765, 7.70708992122614514332932794804, 8.510613124928685313819846009109, 9.902078465133407039524283585983, 11.00481433245061999685969649494, 11.75928771755263924966141257689, 12.44208745103299968706643742150
