| 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
−12.51417166326617039647166131151, −11.87175983670728943751140791745, −10.39198878910555326477265473767, −9.515151681930982053985972745760, −8.806233277198833840942709500842, −7.84019303785907128744972000697, −7.11512662799139724879464277832, −4.56749501541685193103352390718, −3.26699532759527666644938601339, −1.45236796387695968226383264444,
0.19628942847459138995948144343, 2.86382367408066667048745267586, 3.95432596081702197945294914719, 6.49460598053418271931678044529, 7.38948558611182157953194342328, 8.177365688150285367844632967822, 9.047427336689339564261234471175, 10.04965498147514374238761259382, 10.91843593668283242703360012043, 12.34826055344385456108560294626
