L(s) = 1 | + (−3.61 + 1.41i)2-s + (5.16 + 0.600i)3-s + (5.18 − 4.80i)4-s + (−14.1 + 9.62i)5-s + (−19.5 + 5.14i)6-s + (−17.7 + 5.35i)7-s + (1.57 − 3.26i)8-s + (26.2 + 6.19i)9-s + (37.3 − 54.7i)10-s + (4.02 − 26.6i)11-s + (29.6 − 21.6i)12-s + (−20.2 − 16.1i)13-s + (56.4 − 44.4i)14-s + (−78.6 + 41.2i)15-s + (−5.27 + 70.3i)16-s + (25.6 − 7.91i)17-s + ⋯ |
L(s) = 1 | + (−1.27 + 0.501i)2-s + (0.993 + 0.115i)3-s + (0.647 − 0.600i)4-s + (−1.26 + 0.860i)5-s + (−1.32 + 0.350i)6-s + (−0.957 + 0.288i)7-s + (0.0694 − 0.144i)8-s + (0.973 + 0.229i)9-s + (1.18 − 1.73i)10-s + (0.110 − 0.731i)11-s + (0.712 − 0.521i)12-s + (−0.432 − 0.344i)13-s + (1.07 − 0.849i)14-s + (−1.35 + 0.709i)15-s + (−0.0824 + 1.09i)16-s + (0.366 − 0.112i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.191 + 0.981i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.191 + 0.981i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(0.220180 - 0.181384i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.220180 - 0.181384i\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-5.16 - 0.600i)T \) |
| 7 | \( 1 + (17.7 - 5.35i)T \) |
good | 2 | \( 1 + (3.61 - 1.41i)T + (5.86 - 5.44i)T^{2} \) |
| 5 | \( 1 + (14.1 - 9.62i)T + (45.6 - 116. i)T^{2} \) |
| 11 | \( 1 + (-4.02 + 26.6i)T + (-1.27e3 - 392. i)T^{2} \) |
| 13 | \( 1 + (20.2 + 16.1i)T + (488. + 2.14e3i)T^{2} \) |
| 17 | \( 1 + (-25.6 + 7.91i)T + (4.05e3 - 2.76e3i)T^{2} \) |
| 19 | \( 1 + (39.0 + 22.5i)T + (3.42e3 + 5.94e3i)T^{2} \) |
| 23 | \( 1 + (-35.6 + 115. i)T + (-1.00e4 - 6.85e3i)T^{2} \) |
| 29 | \( 1 + (172. + 39.4i)T + (2.19e4 + 1.05e4i)T^{2} \) |
| 31 | \( 1 + (-219. + 126. i)T + (1.48e4 - 2.57e4i)T^{2} \) |
| 37 | \( 1 + (256. + 238. i)T + (3.78e3 + 5.05e4i)T^{2} \) |
| 41 | \( 1 + (-232. - 111. i)T + (4.29e4 + 5.38e4i)T^{2} \) |
| 43 | \( 1 + (279. - 134. i)T + (4.95e4 - 6.21e4i)T^{2} \) |
| 47 | \( 1 + (149. + 380. i)T + (-7.61e4 + 7.06e4i)T^{2} \) |
| 53 | \( 1 + (-225. - 243. i)T + (-1.11e4 + 1.48e5i)T^{2} \) |
| 59 | \( 1 + (656. + 447. i)T + (7.50e4 + 1.91e5i)T^{2} \) |
| 61 | \( 1 + (394. - 425. i)T + (-1.69e4 - 2.26e5i)T^{2} \) |
| 67 | \( 1 + (245. + 425. i)T + (-1.50e5 + 2.60e5i)T^{2} \) |
| 71 | \( 1 + (571. - 130. i)T + (3.22e5 - 1.55e5i)T^{2} \) |
| 73 | \( 1 + (-261. - 102. i)T + (2.85e5 + 2.64e5i)T^{2} \) |
| 79 | \( 1 + (437. - 758. i)T + (-2.46e5 - 4.26e5i)T^{2} \) |
| 83 | \( 1 + (-258. - 324. i)T + (-1.27e5 + 5.57e5i)T^{2} \) |
| 89 | \( 1 + (163. - 24.6i)T + (6.73e5 - 2.07e5i)T^{2} \) |
| 97 | \( 1 + 42.3iT - 9.12e5T^{2} \) |
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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
−12.34826055344385456108560294626, −10.91843593668283242703360012043, −10.04965498147514374238761259382, −9.047427336689339564261234471175, −8.177365688150285367844632967822, −7.38948558611182157953194342328, −6.49460598053418271931678044529, −3.95432596081702197945294914719, −2.86382367408066667048745267586, −0.19628942847459138995948144343,
1.45236796387695968226383264444, 3.26699532759527666644938601339, 4.56749501541685193103352390718, 7.11512662799139724879464277832, 7.84019303785907128744972000697, 8.806233277198833840942709500842, 9.515151681930982053985972745760, 10.39198878910555326477265473767, 11.87175983670728943751140791745, 12.51417166326617039647166131151