| L(s) = 1 | + (0.395 + 0.776i)2-s + (0.296 + 1.87i)3-s + (1.90 − 2.62i)4-s + (−1.33 + 0.970i)6-s + (5.60 − 5.60i)7-s + (6.23 + 0.986i)8-s + (5.14 − 1.67i)9-s + (−4.46 + 13.7i)11-s + (5.47 + 2.78i)12-s + (−5.52 + 10.8i)13-s + (6.57 + 2.13i)14-s + (−2.30 − 7.10i)16-s + (−0.147 + 0.932i)17-s + (3.33 + 3.33i)18-s + (−11.5 − 15.8i)19-s + ⋯ |
| L(s) = 1 | + (0.197 + 0.388i)2-s + (0.0988 + 0.623i)3-s + (0.476 − 0.655i)4-s + (−0.222 + 0.161i)6-s + (0.801 − 0.801i)7-s + (0.778 + 0.123i)8-s + (0.571 − 0.185i)9-s + (−0.406 + 1.25i)11-s + (0.456 + 0.232i)12-s + (−0.425 + 0.834i)13-s + (0.469 + 0.152i)14-s + (−0.144 − 0.443i)16-s + (−0.00869 + 0.0548i)17-s + (0.185 + 0.185i)18-s + (−0.607 − 0.836i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 125 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.877 - 0.479i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 125 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.877 - 0.479i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.84219 + 0.470125i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.84219 + 0.470125i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| good | 2 | \( 1 + (-0.395 - 0.776i)T + (-2.35 + 3.23i)T^{2} \) |
| 3 | \( 1 + (-0.296 - 1.87i)T + (-8.55 + 2.78i)T^{2} \) |
| 7 | \( 1 + (-5.60 + 5.60i)T - 49iT^{2} \) |
| 11 | \( 1 + (4.46 - 13.7i)T + (-97.8 - 71.1i)T^{2} \) |
| 13 | \( 1 + (5.52 - 10.8i)T + (-99.3 - 136. i)T^{2} \) |
| 17 | \( 1 + (0.147 - 0.932i)T + (-274. - 89.3i)T^{2} \) |
| 19 | \( 1 + (11.5 + 15.8i)T + (-111. + 343. i)T^{2} \) |
| 23 | \( 1 + (-4.69 + 2.39i)T + (310. - 427. i)T^{2} \) |
| 29 | \( 1 + (7.87 - 10.8i)T + (-259. - 799. i)T^{2} \) |
| 31 | \( 1 + (-12.1 + 8.83i)T + (296. - 913. i)T^{2} \) |
| 37 | \( 1 + (57.7 + 29.4i)T + (804. + 1.10e3i)T^{2} \) |
| 41 | \( 1 + (16.7 + 51.6i)T + (-1.35e3 + 988. i)T^{2} \) |
| 43 | \( 1 + (3.47 + 3.47i)T + 1.84e3iT^{2} \) |
| 47 | \( 1 + (65.5 - 10.3i)T + (2.10e3 - 682. i)T^{2} \) |
| 53 | \( 1 + (-6.61 - 41.7i)T + (-2.67e3 + 868. i)T^{2} \) |
| 59 | \( 1 + (35.6 - 11.5i)T + (2.81e3 - 2.04e3i)T^{2} \) |
| 61 | \( 1 + (32.4 - 99.9i)T + (-3.01e3 - 2.18e3i)T^{2} \) |
| 67 | \( 1 + (-7.05 + 44.5i)T + (-4.26e3 - 1.38e3i)T^{2} \) |
| 71 | \( 1 + (-36.5 - 26.5i)T + (1.55e3 + 4.79e3i)T^{2} \) |
| 73 | \( 1 + (47.8 - 24.3i)T + (3.13e3 - 4.31e3i)T^{2} \) |
| 79 | \( 1 + (-70.5 + 97.1i)T + (-1.92e3 - 5.93e3i)T^{2} \) |
| 83 | \( 1 + (-60.5 - 9.59i)T + (6.55e3 + 2.12e3i)T^{2} \) |
| 89 | \( 1 + (-48.8 - 15.8i)T + (6.40e3 + 4.65e3i)T^{2} \) |
| 97 | \( 1 + (-36.3 + 5.75i)T + (8.94e3 - 2.90e3i)T^{2} \) |
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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
−13.50230771448280062495259340881, −12.16729876788383679230494493334, −10.84549248454622813363099315374, −10.28736310343261527395458158270, −9.201693890721390459609703950256, −7.44643362295293541752975920573, −6.83542002538437027071960578701, −4.99262413131566284320342836817, −4.33534318477822974760200020468, −1.84003517491243185137310890909,
1.84230761740502082890698041144, 3.21625868164846376604904171143, 5.02903878461147424757313805556, 6.51065778668227000415215420296, 7.956605141959441217064352864308, 8.303971337312040347248041164267, 10.23277470865437127317813073700, 11.25201066122737365950002449272, 12.13352091743506948192666115561, 12.91694488132535182167387417890
