| L(s) = 1 | + (−1.90 − 0.596i)2-s + (−1.09 + 1.33i)3-s + (3.28 + 2.27i)4-s + (−1.04 − 3.45i)5-s + (2.89 − 1.90i)6-s + (−1.78 + 8.95i)7-s + (−4.91 − 6.31i)8-s + (−0.585 − 2.94i)9-s + (−0.0611 + 7.22i)10-s + (7.25 + 0.714i)11-s + (−6.66 + 1.89i)12-s + (−9.36 − 2.84i)13-s + (8.74 − 16.0i)14-s + (5.78 + 2.39i)15-s + (5.61 + 14.9i)16-s + (−1.05 − 2.53i)17-s + ⋯ |
| L(s) = 1 | + (−0.954 − 0.298i)2-s + (−0.366 + 0.446i)3-s + (0.821 + 0.569i)4-s + (−0.209 − 0.691i)5-s + (0.482 − 0.316i)6-s + (−0.254 + 1.27i)7-s + (−0.614 − 0.788i)8-s + (−0.0650 − 0.326i)9-s + (−0.00611 + 0.722i)10-s + (0.659 + 0.0649i)11-s + (−0.555 + 0.158i)12-s + (−0.720 − 0.218i)13-s + (0.624 − 1.14i)14-s + (0.385 + 0.159i)15-s + (0.351 + 0.936i)16-s + (−0.0617 − 0.149i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.837 + 0.547i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.837 + 0.547i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.0454267 - 0.152531i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0454267 - 0.152531i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.90 + 0.596i)T \) |
| 3 | \( 1 + (1.09 - 1.33i)T \) |
| good | 5 | \( 1 + (1.04 + 3.45i)T + (-20.7 + 13.8i)T^{2} \) |
| 7 | \( 1 + (1.78 - 8.95i)T + (-45.2 - 18.7i)T^{2} \) |
| 11 | \( 1 + (-7.25 - 0.714i)T + (118. + 23.6i)T^{2} \) |
| 13 | \( 1 + (9.36 + 2.84i)T + (140. + 93.8i)T^{2} \) |
| 17 | \( 1 + (1.05 + 2.53i)T + (-204. + 204. i)T^{2} \) |
| 19 | \( 1 + (12.9 - 6.91i)T + (200. - 300. i)T^{2} \) |
| 23 | \( 1 + (-2.31 - 1.54i)T + (202. + 488. i)T^{2} \) |
| 29 | \( 1 + (-4.85 + 0.478i)T + (824. - 164. i)T^{2} \) |
| 31 | \( 1 + (-2.60 + 2.60i)T - 961iT^{2} \) |
| 37 | \( 1 + (55.2 + 29.5i)T + (760. + 1.13e3i)T^{2} \) |
| 41 | \( 1 + (28.1 + 18.8i)T + (643. + 1.55e3i)T^{2} \) |
| 43 | \( 1 + (31.4 + 38.3i)T + (-360. + 1.81e3i)T^{2} \) |
| 47 | \( 1 + (13.1 + 31.8i)T + (-1.56e3 + 1.56e3i)T^{2} \) |
| 53 | \( 1 + (86.7 + 8.54i)T + (2.75e3 + 548. i)T^{2} \) |
| 59 | \( 1 + (11.1 + 36.6i)T + (-2.89e3 + 1.93e3i)T^{2} \) |
| 61 | \( 1 + (-7.74 + 9.43i)T + (-725. - 3.64e3i)T^{2} \) |
| 67 | \( 1 + (27.6 + 22.7i)T + (875. + 4.40e3i)T^{2} \) |
| 71 | \( 1 + (63.8 + 12.6i)T + (4.65e3 + 1.92e3i)T^{2} \) |
| 73 | \( 1 + (21.7 - 4.32i)T + (4.92e3 - 2.03e3i)T^{2} \) |
| 79 | \( 1 + (17.8 - 43.0i)T + (-4.41e3 - 4.41e3i)T^{2} \) |
| 83 | \( 1 + (36.4 + 68.1i)T + (-3.82e3 + 5.72e3i)T^{2} \) |
| 89 | \( 1 + (-16.4 - 24.6i)T + (-3.03e3 + 7.31e3i)T^{2} \) |
| 97 | \( 1 + (-2.55 - 2.55i)T + 9.40e3iT^{2} \) |
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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
−10.60990430484970609743341996615, −9.742121106728819072234311139968, −8.907741851540216182331647521176, −8.433354779937144173471207465653, −7.03848359961614876815068900776, −5.99502152469345140220717677284, −4.85637656161938766025995570736, −3.39685478745385275497923918238, −1.96349318116118927952460377750, −0.10137494679533072696257244885,
1.43640966868241160123561098853, 3.10718391112499489217853347538, 4.71383504450064617409304009148, 6.35363993292090055706990743133, 6.88134126890348808298791182565, 7.54385824539525735814901346802, 8.646037196235358800571723809742, 9.822540235368561095830107769464, 10.54017102302937143042855080836, 11.22602658344239557939406512266
