| L(s) = 1 | + (−0.273 + 5.05i)2-s + (−1.98 − 4.80i)3-s + (−17.4 − 1.90i)4-s + (−0.350 + 1.04i)5-s + (24.7 − 8.71i)6-s + (−5.00 − 7.38i)7-s + (7.84 − 47.8i)8-s + (−19.1 + 19.0i)9-s + (−5.16 − 2.05i)10-s + (63.7 − 14.0i)11-s + (25.5 + 87.7i)12-s + (8.42 − 7.15i)13-s + (38.6 − 23.2i)14-s + (5.69 − 0.381i)15-s + (102. + 22.4i)16-s + (43.1 + 29.2i)17-s + ⋯ |
| L(s) = 1 | + (−0.0968 + 1.78i)2-s + (−0.381 − 0.924i)3-s + (−2.18 − 0.237i)4-s + (−0.0313 + 0.0931i)5-s + (1.68 − 0.592i)6-s + (−0.270 − 0.398i)7-s + (0.346 − 2.11i)8-s + (−0.708 + 0.706i)9-s + (−0.163 − 0.0650i)10-s + (1.74 − 0.384i)11-s + (0.615 + 2.11i)12-s + (0.179 − 0.152i)13-s + (0.738 − 0.444i)14-s + (0.0980 − 0.00657i)15-s + (1.59 + 0.351i)16-s + (0.615 + 0.417i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.151 - 0.988i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.151 - 0.988i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.940316 + 0.807177i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.940316 + 0.807177i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (1.98 + 4.80i)T \) |
| 59 | \( 1 + (-421. + 167. i)T \) |
| good | 2 | \( 1 + (0.273 - 5.05i)T + (-7.95 - 0.864i)T^{2} \) |
| 5 | \( 1 + (0.350 - 1.04i)T + (-99.5 - 75.6i)T^{2} \) |
| 7 | \( 1 + (5.00 + 7.38i)T + (-126. + 318. i)T^{2} \) |
| 11 | \( 1 + (-63.7 + 14.0i)T + (1.20e3 - 558. i)T^{2} \) |
| 13 | \( 1 + (-8.42 + 7.15i)T + (355. - 2.16e3i)T^{2} \) |
| 17 | \( 1 + (-43.1 - 29.2i)T + (1.81e3 + 4.56e3i)T^{2} \) |
| 19 | \( 1 + (43.8 - 82.7i)T + (-3.84e3 - 5.67e3i)T^{2} \) |
| 23 | \( 1 + (-25.4 + 24.1i)T + (658. - 1.21e4i)T^{2} \) |
| 29 | \( 1 + (-124. + 6.74i)T + (2.42e4 - 2.63e3i)T^{2} \) |
| 31 | \( 1 + (-115. + 61.1i)T + (1.67e4 - 2.46e4i)T^{2} \) |
| 37 | \( 1 + (18.5 - 3.03i)T + (4.80e4 - 1.61e4i)T^{2} \) |
| 41 | \( 1 + (246. - 260. i)T + (-3.73e3 - 6.88e4i)T^{2} \) |
| 43 | \( 1 + (-69.8 + 317. i)T + (-7.21e4 - 3.33e4i)T^{2} \) |
| 47 | \( 1 + (-438. + 147. i)T + (8.26e4 - 6.28e4i)T^{2} \) |
| 53 | \( 1 + (-234. + 93.2i)T + (1.08e5 - 1.02e5i)T^{2} \) |
| 61 | \( 1 + (518. + 28.0i)T + (2.25e5 + 2.45e4i)T^{2} \) |
| 67 | \( 1 + (-403. - 66.1i)T + (2.85e5 + 9.60e4i)T^{2} \) |
| 71 | \( 1 + (-189. - 563. i)T + (-2.84e5 + 2.16e5i)T^{2} \) |
| 73 | \( 1 + (-352. - 585. i)T + (-1.82e5 + 3.43e5i)T^{2} \) |
| 79 | \( 1 + (220. + 101. i)T + (3.19e5 + 3.75e5i)T^{2} \) |
| 83 | \( 1 + (15.4 + 55.5i)T + (-4.89e5 + 2.94e5i)T^{2} \) |
| 89 | \( 1 + (-34.4 - 634. i)T + (-7.00e5 + 7.62e4i)T^{2} \) |
| 97 | \( 1 + (-65.2 + 108. i)T + (-4.27e5 - 8.06e5i)T^{2} \) |
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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
−12.74176473598346926091131574519, −11.71755327176280095235028069773, −10.24106162367552350124204107787, −8.848105974868999629398651583737, −8.132782417223420134287118012832, −6.93503421048935156222358543060, −6.45601028448039725061913959291, −5.48299744809243618067218107368, −3.91154023705985704447238248030, −0.966730863514025749543628606679,
0.939673452950132170434836070326, 2.80457908106544019240859351699, 3.98280423306467749095112438015, 4.85690543312315834370555416996, 6.49219091377367368658392953128, 8.844209690821652037417063720050, 9.206995464773216342264765918727, 10.18599823413693510923921493249, 11.04380345362441985130133574366, 11.99549600109204538525955735630
