| L(s) = 1 | + (0.747 − 1.85i)2-s + (−2.41 + 1.77i)3-s + (−2.88 − 2.77i)4-s + (0.463 − 2.62i)5-s + (1.48 + 5.81i)6-s + (−4.34 − 5.18i)7-s + (−7.29 + 3.27i)8-s + (2.70 − 8.58i)9-s + (−4.52 − 2.82i)10-s + (−18.5 + 3.26i)11-s + (11.8 + 1.59i)12-s + (−4.76 + 1.73i)13-s + (−12.8 + 4.19i)14-s + (3.53 + 7.17i)15-s + (0.616 + 15.9i)16-s + (7.37 − 12.7i)17-s + ⋯ |
| L(s) = 1 | + (0.373 − 0.927i)2-s + (−0.806 + 0.591i)3-s + (−0.720 − 0.693i)4-s + (0.0926 − 0.525i)5-s + (0.246 + 0.969i)6-s + (−0.621 − 0.740i)7-s + (−0.912 + 0.409i)8-s + (0.300 − 0.953i)9-s + (−0.452 − 0.282i)10-s + (−1.68 + 0.297i)11-s + (0.991 + 0.133i)12-s + (−0.366 + 0.133i)13-s + (−0.918 + 0.299i)14-s + (0.235 + 0.478i)15-s + (0.0385 + 0.999i)16-s + (0.433 − 0.751i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.989 + 0.142i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.989 + 0.142i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.0484333 - 0.677655i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0484333 - 0.677655i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.747 + 1.85i)T \) |
| 3 | \( 1 + (2.41 - 1.77i)T \) |
| good | 5 | \( 1 + (-0.463 + 2.62i)T + (-23.4 - 8.55i)T^{2} \) |
| 7 | \( 1 + (4.34 + 5.18i)T + (-8.50 + 48.2i)T^{2} \) |
| 11 | \( 1 + (18.5 - 3.26i)T + (113. - 41.3i)T^{2} \) |
| 13 | \( 1 + (4.76 - 1.73i)T + (129. - 108. i)T^{2} \) |
| 17 | \( 1 + (-7.37 + 12.7i)T + (-144.5 - 250. i)T^{2} \) |
| 19 | \( 1 + (-28.7 + 16.6i)T + (180.5 - 312. i)T^{2} \) |
| 23 | \( 1 + (-11.3 + 13.5i)T + (-91.8 - 520. i)T^{2} \) |
| 29 | \( 1 + (38.2 + 13.9i)T + (644. + 540. i)T^{2} \) |
| 31 | \( 1 + (4.43 - 5.28i)T + (-166. - 946. i)T^{2} \) |
| 37 | \( 1 + (4.65 - 8.05i)T + (-684.5 - 1.18e3i)T^{2} \) |
| 41 | \( 1 + (20.9 - 7.61i)T + (1.28e3 - 1.08e3i)T^{2} \) |
| 43 | \( 1 + (-42.0 + 7.40i)T + (1.73e3 - 632. i)T^{2} \) |
| 47 | \( 1 + (35.0 + 41.7i)T + (-383. + 2.17e3i)T^{2} \) |
| 53 | \( 1 - 25.6T + 2.80e3T^{2} \) |
| 59 | \( 1 + (15.3 + 2.70i)T + (3.27e3 + 1.19e3i)T^{2} \) |
| 61 | \( 1 + (48.6 - 40.8i)T + (646. - 3.66e3i)T^{2} \) |
| 67 | \( 1 + (11.8 + 32.5i)T + (-3.43e3 + 2.88e3i)T^{2} \) |
| 71 | \( 1 + (23.7 + 13.6i)T + (2.52e3 + 4.36e3i)T^{2} \) |
| 73 | \( 1 + (-18.2 - 31.6i)T + (-2.66e3 + 4.61e3i)T^{2} \) |
| 79 | \( 1 + (-43.9 + 120. i)T + (-4.78e3 - 4.01e3i)T^{2} \) |
| 83 | \( 1 + (-0.482 + 1.32i)T + (-5.27e3 - 4.42e3i)T^{2} \) |
| 89 | \( 1 + (34.1 + 59.2i)T + (-3.96e3 + 6.85e3i)T^{2} \) |
| 97 | \( 1 + (19.7 + 111. i)T + (-8.84e3 + 3.21e3i)T^{2} \) |
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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
−12.90915673545695744227180329192, −11.88669822721422552900298062429, −10.82684294883361077610623996276, −10.01030040655875100947926052935, −9.229874917745587117755276147525, −7.23442410616388588956610719097, −5.42858051918381255522006931274, −4.75991412640755438248964248501, −3.13618675353254584327507760550, −0.48028101826824517897019771804,
3.03752147092226455564404891922, 5.32860533858780992935228681341, 5.85153198873635910197006456525, 7.22120248932709953305405067439, 8.010086716447315646952881820390, 9.663733591204114379315533086346, 10.90496635579853280563084616773, 12.32107076109469712501211359183, 12.87726903266742924736806228736, 13.87426433702542911945698976488
