| L(s) = 1 | + (0.706 + 1.22i)2-s + (−0.275 + 0.106i)3-s + (−1.00 + 1.73i)4-s + (−1.87 + 2.40i)5-s + (−0.324 − 0.262i)6-s + (1.41 − 3.95i)7-s + (−2.82 − 0.00246i)8-s + (−2.15 + 1.95i)9-s + (−4.27 − 0.599i)10-s + (0.720 + 3.20i)11-s + (0.0916 − 0.582i)12-s + (−2.17 + 0.602i)13-s + (5.84 − 1.06i)14-s + (0.261 − 0.861i)15-s + (−1.99 − 3.46i)16-s + (−1.57 + 0.478i)17-s + ⋯ |
| L(s) = 1 | + (0.499 + 0.866i)2-s + (−0.158 + 0.0612i)3-s + (−0.500 + 0.865i)4-s + (−0.839 + 1.07i)5-s + (−0.132 − 0.107i)6-s + (0.535 − 1.49i)7-s + (−0.999 − 0.000871i)8-s + (−0.719 + 0.652i)9-s + (−1.35 − 0.189i)10-s + (0.217 + 0.966i)11-s + (0.0264 − 0.168i)12-s + (−0.604 + 0.167i)13-s + (1.56 − 0.283i)14-s + (0.0674 − 0.222i)15-s + (−0.498 − 0.866i)16-s + (−0.382 + 0.116i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 512 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.888 + 0.458i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 512 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.888 + 0.458i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.181462 - 0.748017i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.181462 - 0.748017i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.706 - 1.22i)T \) |
| good | 3 | \( 1 + (0.275 - 0.106i)T + (2.22 - 2.01i)T^{2} \) |
| 5 | \( 1 + (1.87 - 2.40i)T + (-1.21 - 4.85i)T^{2} \) |
| 7 | \( 1 + (-1.41 + 3.95i)T + (-5.41 - 4.44i)T^{2} \) |
| 11 | \( 1 + (-0.720 - 3.20i)T + (-9.94 + 4.70i)T^{2} \) |
| 13 | \( 1 + (2.17 - 0.602i)T + (11.1 - 6.68i)T^{2} \) |
| 17 | \( 1 + (1.57 - 0.478i)T + (14.1 - 9.44i)T^{2} \) |
| 19 | \( 1 + (0.939 + 1.86i)T + (-11.3 + 15.2i)T^{2} \) |
| 23 | \( 1 + (7.73 - 1.14i)T + (22.0 - 6.67i)T^{2} \) |
| 29 | \( 1 + (-0.547 - 3.15i)T + (-27.3 + 9.76i)T^{2} \) |
| 31 | \( 1 + (-3.41 + 5.10i)T + (-11.8 - 28.6i)T^{2} \) |
| 37 | \( 1 + (-1.19 + 1.02i)T + (5.42 - 36.5i)T^{2} \) |
| 41 | \( 1 + (0.0483 - 0.193i)T + (-36.1 - 19.3i)T^{2} \) |
| 43 | \( 1 + (-3.89 - 8.78i)T + (-28.8 + 31.8i)T^{2} \) |
| 47 | \( 1 + (-6.86 - 8.36i)T + (-9.16 + 46.0i)T^{2} \) |
| 53 | \( 1 + (0.832 - 4.79i)T + (-49.9 - 17.8i)T^{2} \) |
| 59 | \( 1 + (1.54 - 5.59i)T + (-50.6 - 30.3i)T^{2} \) |
| 61 | \( 1 + (-0.0925 - 0.00227i)T + (60.9 + 2.99i)T^{2} \) |
| 67 | \( 1 + (10.5 - 10.0i)T + (3.28 - 66.9i)T^{2} \) |
| 71 | \( 1 + (4.99 - 0.245i)T + (70.6 - 6.95i)T^{2} \) |
| 73 | \( 1 + (-5.33 - 14.9i)T + (-56.4 + 46.3i)T^{2} \) |
| 79 | \( 1 + (-5.62 + 0.553i)T + (77.4 - 15.4i)T^{2} \) |
| 83 | \( 1 + (-3.87 - 3.34i)T + (12.1 + 82.1i)T^{2} \) |
| 89 | \( 1 + (15.6 + 2.31i)T + (85.1 + 25.8i)T^{2} \) |
| 97 | \( 1 + (6.26 - 1.24i)T + (89.6 - 37.1i)T^{2} \) |
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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
−11.38102475071503554636893021995, −10.70621170757740841614647411161, −9.699936291418667656879398453963, −8.218655642601508989766093359833, −7.50218641849175459427597953999, −7.11602139057785774959868326589, −6.00699277442387664350558580668, −4.48649614980197016093516519224, −4.16186943502582620171669875807, −2.68389648851225009658175937973,
0.38115791982076795539506294276, 2.16185394998819439598829461359, 3.46504752612531589094455992215, 4.60141883134682206540319108905, 5.51717220473158181588833521450, 6.18273272277539531715555934526, 8.214824883218150966455810137454, 8.646900064238210018768332941457, 9.366108786826351463005569960896, 10.66693565784076399634923685669
