L(s) = 1 | + (0.5 + 0.866i)2-s + (−0.499 + 0.866i)4-s − 4.13i·5-s + (−2.07 + 1.64i)7-s − 0.999·8-s + (3.57 − 2.06i)10-s + (−1.97 − 3.41i)11-s + (3.54 − 0.675i)13-s + (−2.45 − 0.976i)14-s + (−0.5 − 0.866i)16-s + (2.00 − 3.47i)17-s + (−4.16 + 7.21i)19-s + (3.57 + 2.06i)20-s + (1.97 − 3.41i)22-s + (−5.53 + 3.19i)23-s + ⋯ |
L(s) = 1 | + (0.353 + 0.612i)2-s + (−0.249 + 0.433i)4-s − 1.84i·5-s + (−0.784 + 0.620i)7-s − 0.353·8-s + (1.13 − 0.653i)10-s + (−0.594 − 1.02i)11-s + (0.982 − 0.187i)13-s + (−0.657 − 0.260i)14-s + (−0.125 − 0.216i)16-s + (0.487 − 0.843i)17-s + (−0.956 + 1.65i)19-s + (0.799 + 0.461i)20-s + (0.420 − 0.727i)22-s + (−1.15 + 0.665i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1638 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.915 + 0.401i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1638 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.915 + 0.401i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.3137196091\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.3137196091\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.5 - 0.866i)T \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (2.07 - 1.64i)T \) |
| 13 | \( 1 + (-3.54 + 0.675i)T \) |
good | 5 | \( 1 + 4.13iT - 5T^{2} \) |
| 11 | \( 1 + (1.97 + 3.41i)T + (-5.5 + 9.52i)T^{2} \) |
| 17 | \( 1 + (-2.00 + 3.47i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (4.16 - 7.21i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (5.53 - 3.19i)T + (11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (6.10 - 3.52i)T + (14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 - 4.23T + 31T^{2} \) |
| 37 | \( 1 + (5.40 - 3.11i)T + (18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (2.96 - 1.71i)T + (20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (-2.59 + 4.48i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 - 1.33iT - 47T^{2} \) |
| 53 | \( 1 + 0.335iT - 53T^{2} \) |
| 59 | \( 1 + (-1.89 - 1.09i)T + (29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (10.2 + 5.93i)T + (30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (12.8 - 7.41i)T + (33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (-5.19 + 9.00i)T + (-35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + 2.21T + 73T^{2} \) |
| 79 | \( 1 - 5.68T + 79T^{2} \) |
| 83 | \( 1 + 3.93iT - 83T^{2} \) |
| 89 | \( 1 + (7.32 - 4.23i)T + (44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (-3.21 + 5.56i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−8.788309783176977972503389005598, −8.328836885814606210482317246306, −7.69981984011364181378581023167, −6.19556168747861965351896628106, −5.74758359770279810247990175627, −5.18489675897406226670178668006, −4.01918189175372786771441878428, −3.29126690988140062381739325377, −1.61591918049763317380073582322, −0.10070584083662578953057341733,
2.02772125652149634941885561869, 2.81042130748018186423977233415, 3.73276599383367799285915353410, 4.36387484523181230989885144539, 5.89995177644902832252319589664, 6.49234617073528639620438326815, 7.12672862749608133797417833673, 7.996117641511536070879653463091, 9.234910281656769110155303295115, 10.10413734843542846482387443428