| L(s) = 1 | + (0.281 + 0.959i)2-s + (1.62 − 1.40i)3-s + (−0.841 + 0.540i)4-s + (−1.50 + 1.65i)5-s + (1.80 + 1.15i)6-s + (4.06 + 1.85i)7-s + (−0.755 − 0.654i)8-s + (0.228 − 1.58i)9-s + (−2.01 − 0.978i)10-s + (−2.71 − 0.796i)11-s + (−0.604 + 2.05i)12-s + (4.55 − 2.08i)13-s + (−0.635 + 4.41i)14-s + (−0.116 + 4.79i)15-s + (0.415 − 0.909i)16-s + (0.557 − 0.867i)17-s + ⋯ |
| L(s) = 1 | + (0.199 + 0.678i)2-s + (0.936 − 0.811i)3-s + (−0.420 + 0.270i)4-s + (−0.673 + 0.739i)5-s + (0.736 + 0.473i)6-s + (1.53 + 0.701i)7-s + (−0.267 − 0.231i)8-s + (0.0760 − 0.529i)9-s + (−0.635 − 0.309i)10-s + (−0.817 − 0.240i)11-s + (−0.174 + 0.594i)12-s + (1.26 − 0.577i)13-s + (−0.169 + 1.18i)14-s + (−0.0301 + 1.23i)15-s + (0.103 − 0.227i)16-s + (0.135 − 0.210i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.745 - 0.666i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.745 - 0.666i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.59307 + 0.608690i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.59307 + 0.608690i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.281 - 0.959i)T \) |
| 5 | \( 1 + (1.50 - 1.65i)T \) |
| 23 | \( 1 + (2.84 + 3.86i)T \) |
| good | 3 | \( 1 + (-1.62 + 1.40i)T + (0.426 - 2.96i)T^{2} \) |
| 7 | \( 1 + (-4.06 - 1.85i)T + (4.58 + 5.29i)T^{2} \) |
| 11 | \( 1 + (2.71 + 0.796i)T + (9.25 + 5.94i)T^{2} \) |
| 13 | \( 1 + (-4.55 + 2.08i)T + (8.51 - 9.82i)T^{2} \) |
| 17 | \( 1 + (-0.557 + 0.867i)T + (-7.06 - 15.4i)T^{2} \) |
| 19 | \( 1 + (2.21 - 1.42i)T + (7.89 - 17.2i)T^{2} \) |
| 29 | \( 1 + (6.14 + 3.95i)T + (12.0 + 26.3i)T^{2} \) |
| 31 | \( 1 + (1.21 - 1.39i)T + (-4.41 - 30.6i)T^{2} \) |
| 37 | \( 1 + (5.30 + 0.762i)T + (35.5 + 10.4i)T^{2} \) |
| 41 | \( 1 + (-0.379 - 2.63i)T + (-39.3 + 11.5i)T^{2} \) |
| 43 | \( 1 + (7.97 - 6.91i)T + (6.11 - 42.5i)T^{2} \) |
| 47 | \( 1 + 9.52iT - 47T^{2} \) |
| 53 | \( 1 + (-6.13 - 2.80i)T + (34.7 + 40.0i)T^{2} \) |
| 59 | \( 1 + (3.67 + 8.05i)T + (-38.6 + 44.5i)T^{2} \) |
| 61 | \( 1 + (-0.962 + 1.11i)T + (-8.68 - 60.3i)T^{2} \) |
| 67 | \( 1 + (3.95 + 13.4i)T + (-56.3 + 36.2i)T^{2} \) |
| 71 | \( 1 + (-5.75 + 1.68i)T + (59.7 - 38.3i)T^{2} \) |
| 73 | \( 1 + (-0.315 - 0.491i)T + (-30.3 + 66.4i)T^{2} \) |
| 79 | \( 1 + (-4.90 - 10.7i)T + (-51.7 + 59.7i)T^{2} \) |
| 83 | \( 1 + (-8.64 - 1.24i)T + (79.6 + 23.3i)T^{2} \) |
| 89 | \( 1 + (-0.803 - 0.927i)T + (-12.6 + 88.0i)T^{2} \) |
| 97 | \( 1 + (-5.00 + 0.720i)T + (93.0 - 27.3i)T^{2} \) |
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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
−12.51337403220408486802866124224, −11.42997645428411607413741289493, −10.57437029197761484558059162448, −8.693176475276104190406673816650, −8.084146348291068267987215716977, −7.73033148408401848666430254460, −6.37313787737399837083551760680, −5.12362222001874792808226570681, −3.55162148547310661591857859228, −2.16336862498034394910130724148,
1.68722900064460611567204512196, 3.65407894396793212787755871520, 4.28234771476724477954304684100, 5.29134898117820110628356689669, 7.55173743532048384960957621829, 8.452820410814131318626903826187, 9.016588423075504473123455676318, 10.33742402480995690599417675418, 11.08301092260983663820696099504, 11.89768302791739454898845795185
