| L(s) = 1 | + 0.381·2-s − 1.85·4-s + 3.61·5-s + 0.236·7-s − 1.47·8-s + 1.38·10-s + 6.23·11-s − 3.85·13-s + 0.0901·14-s + 3.14·16-s − 0.236·17-s − 1.76·19-s − 6.70·20-s + 2.38·22-s − 1.85·23-s + 8.09·25-s − 1.47·26-s − 0.437·28-s + 6.32·29-s + 8.47·31-s + 4.14·32-s − 0.0901·34-s + 0.854·35-s + 5.09·37-s − 0.673·38-s − 5.32·40-s + 9.61·41-s + ⋯ |
| L(s) = 1 | + 0.270·2-s − 0.927·4-s + 1.61·5-s + 0.0892·7-s − 0.520·8-s + 0.437·10-s + 1.88·11-s − 1.06·13-s + 0.0240·14-s + 0.786·16-s − 0.0572·17-s − 0.404·19-s − 1.49·20-s + 0.507·22-s − 0.386·23-s + 1.61·25-s − 0.288·26-s − 0.0827·28-s + 1.17·29-s + 1.52·31-s + 0.732·32-s − 0.0154·34-s + 0.144·35-s + 0.836·37-s − 0.109·38-s − 0.842·40-s + 1.50·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 639 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 639 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.904704597\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.904704597\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 71 | \( 1 + T \) |
| good | 2 | \( 1 - 0.381T + 2T^{2} \) |
| 5 | \( 1 - 3.61T + 5T^{2} \) |
| 7 | \( 1 - 0.236T + 7T^{2} \) |
| 11 | \( 1 - 6.23T + 11T^{2} \) |
| 13 | \( 1 + 3.85T + 13T^{2} \) |
| 17 | \( 1 + 0.236T + 17T^{2} \) |
| 19 | \( 1 + 1.76T + 19T^{2} \) |
| 23 | \( 1 + 1.85T + 23T^{2} \) |
| 29 | \( 1 - 6.32T + 29T^{2} \) |
| 31 | \( 1 - 8.47T + 31T^{2} \) |
| 37 | \( 1 - 5.09T + 37T^{2} \) |
| 41 | \( 1 - 9.61T + 41T^{2} \) |
| 43 | \( 1 + 4.14T + 43T^{2} \) |
| 47 | \( 1 + 10.8T + 47T^{2} \) |
| 53 | \( 1 + 5.61T + 53T^{2} \) |
| 59 | \( 1 - 9.18T + 59T^{2} \) |
| 61 | \( 1 - 5T + 61T^{2} \) |
| 67 | \( 1 + 14.7T + 67T^{2} \) |
| 73 | \( 1 + 1.23T + 73T^{2} \) |
| 79 | \( 1 + 3.61T + 79T^{2} \) |
| 83 | \( 1 - 3.76T + 83T^{2} \) |
| 89 | \( 1 - 2.52T + 89T^{2} \) |
| 97 | \( 1 + 5.32T + 97T^{2} \) |
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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
−10.15777899817420808132965582332, −9.688384051352312647275310430231, −9.109216760560140195489544560494, −8.178197509404552876456551457388, −6.59727717204065338929515832589, −6.13598592227650228070662059671, −4.98474986847994816398678689241, −4.23489527120816658066913111390, −2.74387013959626747897704019665, −1.33103603999675656270187835880,
1.33103603999675656270187835880, 2.74387013959626747897704019665, 4.23489527120816658066913111390, 4.98474986847994816398678689241, 6.13598592227650228070662059671, 6.59727717204065338929515832589, 8.178197509404552876456551457388, 9.109216760560140195489544560494, 9.688384051352312647275310430231, 10.15777899817420808132965582332
