| L(s) = 1 | + (−2.03 + 0.841i)2-s + (−0.0315 + 0.158i)3-s + (0.590 − 0.590i)4-s + (4.46 − 2.98i)5-s + (−0.0693 − 0.348i)6-s + (−5.19 − 3.46i)7-s + (2.66 − 6.42i)8-s + (8.29 + 3.43i)9-s + (−6.55 + 9.80i)10-s + (−14.3 + 2.84i)11-s + (0.0749 + 0.112i)12-s + (−4.79 − 4.79i)13-s + (13.4 + 2.67i)14-s + (0.331 + 0.801i)15-s + 18.6i·16-s + ⋯ |
| L(s) = 1 | + (−1.01 + 0.420i)2-s + (−0.0105 + 0.0528i)3-s + (0.147 − 0.147i)4-s + (0.892 − 0.596i)5-s + (−0.0115 − 0.0580i)6-s + (−0.741 − 0.495i)7-s + (0.332 − 0.803i)8-s + (0.921 + 0.381i)9-s + (−0.655 + 0.980i)10-s + (−1.30 + 0.259i)11-s + (0.00624 + 0.00935i)12-s + (−0.369 − 0.369i)13-s + (0.961 + 0.191i)14-s + (0.0221 + 0.0534i)15-s + 1.16i·16-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 289 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.563 + 0.825i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 289 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.563 + 0.825i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.144463 - 0.273590i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.144463 - 0.273590i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 17 | \( 1 \) |
| good | 2 | \( 1 + (2.03 - 0.841i)T + (2.82 - 2.82i)T^{2} \) |
| 3 | \( 1 + (0.0315 - 0.158i)T + (-8.31 - 3.44i)T^{2} \) |
| 5 | \( 1 + (-4.46 + 2.98i)T + (9.56 - 23.0i)T^{2} \) |
| 7 | \( 1 + (5.19 + 3.46i)T + (18.7 + 45.2i)T^{2} \) |
| 11 | \( 1 + (14.3 - 2.84i)T + (111. - 46.3i)T^{2} \) |
| 13 | \( 1 + (4.79 + 4.79i)T + 169iT^{2} \) |
| 19 | \( 1 + (23.0 - 9.56i)T + (255. - 255. i)T^{2} \) |
| 23 | \( 1 + (2.55 + 12.8i)T + (-488. + 202. i)T^{2} \) |
| 29 | \( 1 + (18.3 + 27.4i)T + (-321. + 776. i)T^{2} \) |
| 31 | \( 1 + (1.19 + 0.236i)T + (887. + 367. i)T^{2} \) |
| 37 | \( 1 + (-9.06 + 45.5i)T + (-1.26e3 - 523. i)T^{2} \) |
| 41 | \( 1 + (26.1 + 17.4i)T + (643. + 1.55e3i)T^{2} \) |
| 43 | \( 1 + (67.3 + 27.8i)T + (1.30e3 + 1.30e3i)T^{2} \) |
| 47 | \( 1 + (-10.4 - 10.4i)T + 2.20e3iT^{2} \) |
| 53 | \( 1 + (4.77 - 1.97i)T + (1.98e3 - 1.98e3i)T^{2} \) |
| 59 | \( 1 + (10.8 - 26.1i)T + (-2.46e3 - 2.46e3i)T^{2} \) |
| 61 | \( 1 + (-45.9 + 68.7i)T + (-1.42e3 - 3.43e3i)T^{2} \) |
| 67 | \( 1 + 44.5iT - 4.48e3T^{2} \) |
| 71 | \( 1 + (11.3 - 56.8i)T + (-4.65e3 - 1.92e3i)T^{2} \) |
| 73 | \( 1 + (-1.12 + 0.748i)T + (2.03e3 - 4.92e3i)T^{2} \) |
| 79 | \( 1 + (29.1 - 5.79i)T + (5.76e3 - 2.38e3i)T^{2} \) |
| 83 | \( 1 + (-25.7 - 62.2i)T + (-4.87e3 + 4.87e3i)T^{2} \) |
| 89 | \( 1 + (90.1 - 90.1i)T - 7.92e3iT^{2} \) |
| 97 | \( 1 + (-37.5 - 56.2i)T + (-3.60e3 + 8.69e3i)T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.59167411423257936238936384552, −10.10412187888590461554363932417, −9.508225206100201261288549828209, −8.359273050852008735171578192919, −7.54184406164818006168808878676, −6.57942517690638006461582698548, −5.27838431449110748652871355936, −4.01955780485695659819905020431, −2.01158574181271019883737906348, −0.20235573671091793538666257315,
1.82934419487113979986571206166, 2.91589628188578422142376540143, 4.90184423858708251830062118151, 6.11811447242724363478465642988, 7.10465238445896123630121678258, 8.351916639943272566191970911851, 9.350810769320987102650934609811, 10.04443505142521204576043852901, 10.51389303420935611656516412554, 11.63646491519529447745490079083
