| L(s) = 1 | + (2.47 + 0.971i)2-s + (−4.78 + 2.02i)3-s + (−0.685 − 0.635i)4-s + (−1.93 − 1.31i)5-s + (−13.8 + 0.355i)6-s + (18.3 − 2.32i)7-s + (−10.3 − 21.3i)8-s + (18.8 − 19.3i)9-s + (−3.50 − 5.13i)10-s + (−1.51 − 10.0i)11-s + (4.56 + 1.65i)12-s + (20.5 − 16.3i)13-s + (47.7 + 12.0i)14-s + (11.9 + 2.39i)15-s + (−4.15 − 55.4i)16-s + (50.7 + 15.6i)17-s + ⋯ |
| L(s) = 1 | + (0.874 + 0.343i)2-s + (−0.921 + 0.389i)3-s + (−0.0856 − 0.0794i)4-s + (−0.172 − 0.117i)5-s + (−0.939 + 0.0241i)6-s + (0.992 − 0.125i)7-s + (−0.455 − 0.945i)8-s + (0.697 − 0.716i)9-s + (−0.110 − 0.162i)10-s + (−0.0414 − 0.275i)11-s + (0.109 + 0.0398i)12-s + (0.438 − 0.349i)13-s + (0.911 + 0.230i)14-s + (0.205 + 0.0412i)15-s + (−0.0649 − 0.867i)16-s + (0.724 + 0.223i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.801 + 0.598i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.801 + 0.598i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.67768 - 0.557193i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.67768 - 0.557193i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (4.78 - 2.02i)T \) |
| 7 | \( 1 + (-18.3 + 2.32i)T \) |
| good | 2 | \( 1 + (-2.47 - 0.971i)T + (5.86 + 5.44i)T^{2} \) |
| 5 | \( 1 + (1.93 + 1.31i)T + (45.6 + 116. i)T^{2} \) |
| 11 | \( 1 + (1.51 + 10.0i)T + (-1.27e3 + 392. i)T^{2} \) |
| 13 | \( 1 + (-20.5 + 16.3i)T + (488. - 2.14e3i)T^{2} \) |
| 17 | \( 1 + (-50.7 - 15.6i)T + (4.05e3 + 2.76e3i)T^{2} \) |
| 19 | \( 1 + (-92.1 + 53.1i)T + (3.42e3 - 5.94e3i)T^{2} \) |
| 23 | \( 1 + (26.5 + 86.1i)T + (-1.00e4 + 6.85e3i)T^{2} \) |
| 29 | \( 1 + (12.1 - 2.76i)T + (2.19e4 - 1.05e4i)T^{2} \) |
| 31 | \( 1 + (36.5 + 21.1i)T + (1.48e4 + 2.57e4i)T^{2} \) |
| 37 | \( 1 + (-50.7 + 47.1i)T + (3.78e3 - 5.05e4i)T^{2} \) |
| 41 | \( 1 + (285. - 137. i)T + (4.29e4 - 5.38e4i)T^{2} \) |
| 43 | \( 1 + (218. + 105. i)T + (4.95e4 + 6.21e4i)T^{2} \) |
| 47 | \( 1 + (-9.58 + 24.4i)T + (-7.61e4 - 7.06e4i)T^{2} \) |
| 53 | \( 1 + (319. - 344. i)T + (-1.11e4 - 1.48e5i)T^{2} \) |
| 59 | \( 1 + (-502. + 342. i)T + (7.50e4 - 1.91e5i)T^{2} \) |
| 61 | \( 1 + (-225. - 242. i)T + (-1.69e4 + 2.26e5i)T^{2} \) |
| 67 | \( 1 + (-285. + 494. i)T + (-1.50e5 - 2.60e5i)T^{2} \) |
| 71 | \( 1 + (-360. - 82.2i)T + (3.22e5 + 1.55e5i)T^{2} \) |
| 73 | \( 1 + (-932. + 365. i)T + (2.85e5 - 2.64e5i)T^{2} \) |
| 79 | \( 1 + (-134. - 232. i)T + (-2.46e5 + 4.26e5i)T^{2} \) |
| 83 | \( 1 + (896. - 1.12e3i)T + (-1.27e5 - 5.57e5i)T^{2} \) |
| 89 | \( 1 + (-108. - 16.3i)T + (6.73e5 + 2.07e5i)T^{2} \) |
| 97 | \( 1 - 352. iT - 9.12e5T^{2} \) |
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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
−12.45296145414965444066780578983, −11.64170638205521244964833836125, −10.61722226542650876119185217296, −9.600749557618468092442209261057, −8.142993725787478175339298587842, −6.70440038248412794787208930587, −5.56388408942067447249795507362, −4.84198051617821092753320351055, −3.71121257587923065029932091388, −0.820971244800549806022984822101,
1.65615572163420365436631205910, 3.64448822592792739233264586504, 4.99194995148298844586402670686, 5.67372386264382880509884781649, 7.30931700527876842119260392384, 8.264337105158118254483361126658, 9.873779332034285634903062266795, 11.38639118464230795936454546193, 11.60074203176022074555579834028, 12.56711395065698871725580703616
