| L(s) = 1 | + (−2.65 + 1.53i)2-s + (1.23 − 5.04i)3-s + (0.685 − 1.18i)4-s + 5.50·5-s + (4.44 + 15.2i)6-s + (−18.4 + 1.02i)7-s − 20.2i·8-s + (−23.9 − 12.5i)9-s + (−14.5 + 8.42i)10-s − 59.4i·11-s + (−5.14 − 4.93i)12-s + (12.7 − 7.35i)13-s + (47.4 − 31.0i)14-s + (6.81 − 27.7i)15-s + (36.5 + 63.2i)16-s + (−29.3 − 50.7i)17-s + ⋯ |
| L(s) = 1 | + (−0.937 + 0.541i)2-s + (0.238 − 0.971i)3-s + (0.0857 − 0.148i)4-s + 0.492·5-s + (0.302 + 1.03i)6-s + (−0.998 + 0.0556i)7-s − 0.896i·8-s + (−0.886 − 0.462i)9-s + (−0.461 + 0.266i)10-s − 1.62i·11-s + (−0.123 − 0.118i)12-s + (0.271 − 0.157i)13-s + (0.905 − 0.592i)14-s + (0.117 − 0.477i)15-s + (0.571 + 0.989i)16-s + (−0.418 − 0.724i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.202 + 0.979i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.202 + 0.979i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.390534 - 0.479309i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.390534 - 0.479309i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-1.23 + 5.04i)T \) |
| 7 | \( 1 + (18.4 - 1.02i)T \) |
| good | 2 | \( 1 + (2.65 - 1.53i)T + (4 - 6.92i)T^{2} \) |
| 5 | \( 1 - 5.50T + 125T^{2} \) |
| 11 | \( 1 + 59.4iT - 1.33e3T^{2} \) |
| 13 | \( 1 + (-12.7 + 7.35i)T + (1.09e3 - 1.90e3i)T^{2} \) |
| 17 | \( 1 + (29.3 + 50.7i)T + (-2.45e3 + 4.25e3i)T^{2} \) |
| 19 | \( 1 + (66.1 + 38.2i)T + (3.42e3 + 5.94e3i)T^{2} \) |
| 23 | \( 1 + 26.0iT - 1.21e4T^{2} \) |
| 29 | \( 1 + (-217. - 125. i)T + (1.21e4 + 2.11e4i)T^{2} \) |
| 31 | \( 1 + (-190. - 109. i)T + (1.48e4 + 2.57e4i)T^{2} \) |
| 37 | \( 1 + (97.2 - 168. i)T + (-2.53e4 - 4.38e4i)T^{2} \) |
| 41 | \( 1 + (-37.6 - 65.2i)T + (-3.44e4 + 5.96e4i)T^{2} \) |
| 43 | \( 1 + (-210. + 365. i)T + (-3.97e4 - 6.88e4i)T^{2} \) |
| 47 | \( 1 + (191. + 331. i)T + (-5.19e4 + 8.99e4i)T^{2} \) |
| 53 | \( 1 + (-92.7 + 53.5i)T + (7.44e4 - 1.28e5i)T^{2} \) |
| 59 | \( 1 + (266. - 462. i)T + (-1.02e5 - 1.77e5i)T^{2} \) |
| 61 | \( 1 + (-35.7 + 20.6i)T + (1.13e5 - 1.96e5i)T^{2} \) |
| 67 | \( 1 + (166. - 288. i)T + (-1.50e5 - 2.60e5i)T^{2} \) |
| 71 | \( 1 + 500. iT - 3.57e5T^{2} \) |
| 73 | \( 1 + (-351. + 203. i)T + (1.94e5 - 3.36e5i)T^{2} \) |
| 79 | \( 1 + (75.1 + 130. i)T + (-2.46e5 + 4.26e5i)T^{2} \) |
| 83 | \( 1 + (-493. + 854. i)T + (-2.85e5 - 4.95e5i)T^{2} \) |
| 89 | \( 1 + (-311. + 539. i)T + (-3.52e5 - 6.10e5i)T^{2} \) |
| 97 | \( 1 + (407. + 235. i)T + (4.56e5 + 7.90e5i)T^{2} \) |
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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
−13.79262385706498239691352557525, −13.31263153452409587874434728199, −12.06396153450887190036842225848, −10.42347555537908911632636608099, −8.997156889066932904666951217251, −8.402388382387631361862557590799, −6.87027252447940456196437070869, −6.11791019035169092816576634640, −3.09790546437780686717919175268, −0.54698119297571750728778228218,
2.30481901863789499742088055247, 4.36852927766733322101223516971, 6.13743382906934340866160001232, 8.178192280337683833206312524567, 9.535749221695246307441949843676, 9.866311783867098424550985793607, 10.84576892092090042982364601638, 12.35056457191299817433706078651, 13.76531236512395691744900908854, 14.93233559046709418948712496735
