| L(s) = 1 | + (−0.103 + 1.99i)2-s + (−3.97 − 0.414i)4-s + (−1.50 + 8.54i)5-s + (7.53 + 8.98i)7-s + (1.24 − 7.90i)8-s + (−16.9 − 3.89i)10-s + (2.27 − 0.400i)11-s + (−4.42 + 1.61i)13-s + (−18.7 + 14.1i)14-s + (15.6 + 3.30i)16-s + (−6.77 + 11.7i)17-s + (13.9 − 8.02i)19-s + (9.53 − 33.3i)20-s + (0.564 + 4.57i)22-s + (3.20 − 3.82i)23-s + ⋯ |
| L(s) = 1 | + (−0.0519 + 0.998i)2-s + (−0.994 − 0.103i)4-s + (−0.301 + 1.70i)5-s + (1.07 + 1.28i)7-s + (0.155 − 0.987i)8-s + (−1.69 − 0.389i)10-s + (0.206 − 0.0364i)11-s + (−0.340 + 0.123i)13-s + (−1.33 + 1.00i)14-s + (0.978 + 0.206i)16-s + (−0.398 + 0.690i)17-s + (0.731 − 0.422i)19-s + (0.476 − 1.66i)20-s + (0.0256 + 0.208i)22-s + (0.139 − 0.166i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.942 + 0.334i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.942 + 0.334i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.218347 - 1.26663i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.218347 - 1.26663i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.103 - 1.99i)T \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 + (1.50 - 8.54i)T + (-23.4 - 8.55i)T^{2} \) |
| 7 | \( 1 + (-7.53 - 8.98i)T + (-8.50 + 48.2i)T^{2} \) |
| 11 | \( 1 + (-2.27 + 0.400i)T + (113. - 41.3i)T^{2} \) |
| 13 | \( 1 + (4.42 - 1.61i)T + (129. - 108. i)T^{2} \) |
| 17 | \( 1 + (6.77 - 11.7i)T + (-144.5 - 250. i)T^{2} \) |
| 19 | \( 1 + (-13.9 + 8.02i)T + (180.5 - 312. i)T^{2} \) |
| 23 | \( 1 + (-3.20 + 3.82i)T + (-91.8 - 520. i)T^{2} \) |
| 29 | \( 1 + (34.0 + 12.4i)T + (644. + 540. i)T^{2} \) |
| 31 | \( 1 + (0.283 - 0.337i)T + (-166. - 946. i)T^{2} \) |
| 37 | \( 1 + (-18.9 + 32.7i)T + (-684.5 - 1.18e3i)T^{2} \) |
| 41 | \( 1 + (2.35 - 0.858i)T + (1.28e3 - 1.08e3i)T^{2} \) |
| 43 | \( 1 + (17.5 - 3.10i)T + (1.73e3 - 632. i)T^{2} \) |
| 47 | \( 1 + (5.64 + 6.72i)T + (-383. + 2.17e3i)T^{2} \) |
| 53 | \( 1 - 41.4T + 2.80e3T^{2} \) |
| 59 | \( 1 + (25.5 + 4.51i)T + (3.27e3 + 1.19e3i)T^{2} \) |
| 61 | \( 1 + (-51.1 + 42.9i)T + (646. - 3.66e3i)T^{2} \) |
| 67 | \( 1 + (-32.3 - 88.9i)T + (-3.43e3 + 2.88e3i)T^{2} \) |
| 71 | \( 1 + (-100. - 57.8i)T + (2.52e3 + 4.36e3i)T^{2} \) |
| 73 | \( 1 + (-22.5 - 39.0i)T + (-2.66e3 + 4.61e3i)T^{2} \) |
| 79 | \( 1 + (19.8 - 54.4i)T + (-4.78e3 - 4.01e3i)T^{2} \) |
| 83 | \( 1 + (-22.8 + 62.8i)T + (-5.27e3 - 4.42e3i)T^{2} \) |
| 89 | \( 1 + (54.3 + 94.0i)T + (-3.96e3 + 6.85e3i)T^{2} \) |
| 97 | \( 1 + (-30.2 - 171. i)T + (-8.84e3 + 3.21e3i)T^{2} \) |
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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
−11.68178568716439216311014608500, −11.09126460702676269629256811905, −9.921980541392919734618192106551, −8.887807407968262595308287173166, −7.927362007016493071630935809770, −7.13465048196700598420218716129, −6.16688108202707669091073082925, −5.23197766269475183507176005836, −3.82460831060074089616308290930, −2.33741785719996156660576366502,
0.66616770695104849771725135464, 1.63010039691784342792287303251, 3.74395789396249147808067533302, 4.66885101348399955693723325880, 5.24037525964289389257402387318, 7.48273901146560140803227174313, 8.180503405606676683566785831888, 9.131403561894545084138028897104, 9.935002771416188124721790668125, 11.10770674920668699338855998000
