| L(s) = 1 | + (5.10 + 2.00i)2-s + (−2.68 + 4.44i)3-s + (16.1 + 15.0i)4-s + (8.40 + 5.72i)5-s + (−22.6 + 17.3i)6-s + (9.18 − 16.0i)7-s + (33.5 + 69.6i)8-s + (−12.5 − 23.9i)9-s + (31.4 + 46.1i)10-s + (−10.5 − 69.9i)11-s + (−110. + 31.6i)12-s + (−35.9 + 28.6i)13-s + (79.1 − 63.7i)14-s + (−48.0 + 21.9i)15-s + (18.5 + 246. i)16-s + (−28.4 − 8.78i)17-s + ⋯ |
| L(s) = 1 | + (1.80 + 0.708i)2-s + (−0.517 + 0.855i)3-s + (2.02 + 1.87i)4-s + (0.751 + 0.512i)5-s + (−1.54 + 1.17i)6-s + (0.495 − 0.868i)7-s + (1.48 + 3.07i)8-s + (−0.464 − 0.885i)9-s + (0.993 + 1.45i)10-s + (−0.289 − 1.91i)11-s + (−2.65 + 0.760i)12-s + (−0.767 + 0.612i)13-s + (1.51 − 1.21i)14-s + (−0.827 + 0.378i)15-s + (0.289 + 3.85i)16-s + (−0.406 − 0.125i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.263 - 0.964i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.263 - 0.964i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.67661 + 3.50744i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.67661 + 3.50744i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (2.68 - 4.44i)T \) |
| 7 | \( 1 + (-9.18 + 16.0i)T \) |
| good | 2 | \( 1 + (-5.10 - 2.00i)T + (5.86 + 5.44i)T^{2} \) |
| 5 | \( 1 + (-8.40 - 5.72i)T + (45.6 + 116. i)T^{2} \) |
| 11 | \( 1 + (10.5 + 69.9i)T + (-1.27e3 + 392. i)T^{2} \) |
| 13 | \( 1 + (35.9 - 28.6i)T + (488. - 2.14e3i)T^{2} \) |
| 17 | \( 1 + (28.4 + 8.78i)T + (4.05e3 + 2.76e3i)T^{2} \) |
| 19 | \( 1 + (-2.68 + 1.55i)T + (3.42e3 - 5.94e3i)T^{2} \) |
| 23 | \( 1 + (2.31 + 7.49i)T + (-1.00e4 + 6.85e3i)T^{2} \) |
| 29 | \( 1 + (-22.3 + 5.09i)T + (2.19e4 - 1.05e4i)T^{2} \) |
| 31 | \( 1 + (-98.9 - 57.1i)T + (1.48e4 + 2.57e4i)T^{2} \) |
| 37 | \( 1 + (-16.9 + 15.7i)T + (3.78e3 - 5.05e4i)T^{2} \) |
| 41 | \( 1 + (291. - 140. i)T + (4.29e4 - 5.38e4i)T^{2} \) |
| 43 | \( 1 + (-77.6 - 37.4i)T + (4.95e4 + 6.21e4i)T^{2} \) |
| 47 | \( 1 + (-49.2 + 125. i)T + (-7.61e4 - 7.06e4i)T^{2} \) |
| 53 | \( 1 + (-228. + 246. i)T + (-1.11e4 - 1.48e5i)T^{2} \) |
| 59 | \( 1 + (564. - 385. i)T + (7.50e4 - 1.91e5i)T^{2} \) |
| 61 | \( 1 + (-397. - 428. i)T + (-1.69e4 + 2.26e5i)T^{2} \) |
| 67 | \( 1 + (286. - 496. i)T + (-1.50e5 - 2.60e5i)T^{2} \) |
| 71 | \( 1 + (-428. - 97.6i)T + (3.22e5 + 1.55e5i)T^{2} \) |
| 73 | \( 1 + (174. - 68.4i)T + (2.85e5 - 2.64e5i)T^{2} \) |
| 79 | \( 1 + (-478. - 828. i)T + (-2.46e5 + 4.26e5i)T^{2} \) |
| 83 | \( 1 + (-305. + 382. i)T + (-1.27e5 - 5.57e5i)T^{2} \) |
| 89 | \( 1 + (719. + 108. i)T + (6.73e5 + 2.07e5i)T^{2} \) |
| 97 | \( 1 + 706. iT - 9.12e5T^{2} \) |
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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
−13.35948081100001303943305548107, −11.85523020568359521069454647425, −11.17010171021872800427358223377, −10.30886751598245442919614776659, −8.413859477314571650092816017431, −6.92500835114953011174240627924, −6.07181186910281031357213133909, −5.13469219350246603133270171642, −4.07104732337133339792686840659, −2.86463990032999875122411127613,
1.73255382668928543063704250554, 2.43101854447695864143025986630, 4.79877611970028155777107258152, 5.26010313912253130908776819907, 6.36284662785296915374469057594, 7.54538778476422516460355025442, 9.659204898115140390571425663886, 10.69622602357860115187197862322, 11.92681175244679502057218088536, 12.40618913583326814516995780664
