| L(s) = 1 | + (−0.419 + 0.136i)2-s + (1.18 + 1.62i)3-s + (−1.46 + 1.06i)4-s + (−0.544 − 2.16i)5-s + (−0.719 − 0.522i)6-s − 3.59i·7-s + (0.987 − 1.35i)8-s + (−0.326 + 1.00i)9-s + (0.524 + 0.836i)10-s + (−1.50 − 4.64i)11-s + (−3.45 − 1.12i)12-s + (5.02 + 1.63i)13-s + (0.490 + 1.50i)14-s + (2.88 − 3.45i)15-s + (0.886 − 2.72i)16-s + (−0.587 + 0.809i)17-s + ⋯ |
| L(s) = 1 | + (−0.296 + 0.0964i)2-s + (0.683 + 0.940i)3-s + (−0.730 + 0.530i)4-s + (−0.243 − 0.969i)5-s + (−0.293 − 0.213i)6-s − 1.35i·7-s + (0.349 − 0.480i)8-s + (−0.108 + 0.335i)9-s + (0.165 + 0.264i)10-s + (−0.455 − 1.40i)11-s + (−0.998 − 0.324i)12-s + (1.39 + 0.452i)13-s + (0.131 + 0.403i)14-s + (0.746 − 0.892i)15-s + (0.221 − 0.682i)16-s + (−0.142 + 0.196i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 425 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.763 + 0.645i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 425 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.763 + 0.645i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.03552 - 0.379079i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.03552 - 0.379079i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + (0.544 + 2.16i)T \) |
| 17 | \( 1 + (0.587 - 0.809i)T \) |
| good | 2 | \( 1 + (0.419 - 0.136i)T + (1.61 - 1.17i)T^{2} \) |
| 3 | \( 1 + (-1.18 - 1.62i)T + (-0.927 + 2.85i)T^{2} \) |
| 7 | \( 1 + 3.59iT - 7T^{2} \) |
| 11 | \( 1 + (1.50 + 4.64i)T + (-8.89 + 6.46i)T^{2} \) |
| 13 | \( 1 + (-5.02 - 1.63i)T + (10.5 + 7.64i)T^{2} \) |
| 19 | \( 1 + (2.39 + 1.73i)T + (5.87 + 18.0i)T^{2} \) |
| 23 | \( 1 + (5.79 - 1.88i)T + (18.6 - 13.5i)T^{2} \) |
| 29 | \( 1 + (-2.97 + 2.15i)T + (8.96 - 27.5i)T^{2} \) |
| 31 | \( 1 + (-4.70 - 3.41i)T + (9.57 + 29.4i)T^{2} \) |
| 37 | \( 1 + (-3.66 - 1.19i)T + (29.9 + 21.7i)T^{2} \) |
| 41 | \( 1 + (1.71 - 5.28i)T + (-33.1 - 24.0i)T^{2} \) |
| 43 | \( 1 + 12.3iT - 43T^{2} \) |
| 47 | \( 1 + (0.643 + 0.885i)T + (-14.5 + 44.6i)T^{2} \) |
| 53 | \( 1 + (7.65 + 10.5i)T + (-16.3 + 50.4i)T^{2} \) |
| 59 | \( 1 + (1.53 - 4.72i)T + (-47.7 - 34.6i)T^{2} \) |
| 61 | \( 1 + (-2.19 - 6.76i)T + (-49.3 + 35.8i)T^{2} \) |
| 67 | \( 1 + (-5.88 + 8.10i)T + (-20.7 - 63.7i)T^{2} \) |
| 71 | \( 1 + (-13.0 + 9.51i)T + (21.9 - 67.5i)T^{2} \) |
| 73 | \( 1 + (4.59 - 1.49i)T + (59.0 - 42.9i)T^{2} \) |
| 79 | \( 1 + (1.22 - 0.891i)T + (24.4 - 75.1i)T^{2} \) |
| 83 | \( 1 + (-3.33 + 4.59i)T + (-25.6 - 78.9i)T^{2} \) |
| 89 | \( 1 + (-0.475 - 1.46i)T + (-72.0 + 52.3i)T^{2} \) |
| 97 | \( 1 + (-4.79 - 6.59i)T + (-29.9 + 92.2i)T^{2} \) |
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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
−10.78947288655203558727859756074, −10.02778315974813642828329964475, −9.080389485957809482107037261153, −8.390898482060950543718584787607, −8.003936388471896226920487703071, −6.41162915331697244427640248421, −4.84357383945784612928334059892, −3.93612761326849577108548319937, −3.55204540372966587493722482985, −0.77606971853239199896804059964,
1.82942515340136475293102655657, 2.72776223794179597662032942971, 4.37006978355466204753178249415, 5.79404042932313867709150971609, 6.61306592607471793794984451734, 8.031929242867945469287622031183, 8.236215932505869029750301702599, 9.436984434855237753228980842734, 10.26531975954252568076383400025, 11.17749637643534124784612423141
