| 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
−11.89768302791739454898845795185, −11.08301092260983663820696099504, −10.33742402480995690599417675418, −9.016588423075504473123455676318, −8.452820410814131318626903826187, −7.55173743532048384960957621829, −5.29134898117820110628356689669, −4.28234771476724477954304684100, −3.65407894396793212787755871520, −1.68722900064460611567204512196,
2.16336862498034394910130724148, 3.55162148547310661591857859228, 5.12362222001874792808226570681, 6.37313787737399837083551760680, 7.73033148408401848666430254460, 8.084146348291068267987215716977, 8.693176475276104190406673816650, 10.57437029197761484558059162448, 11.42997645428411607413741289493, 12.51337403220408486802866124224
