| L(s) = 1 | + (−0.236 − 0.517i)2-s + (−0.959 + 0.281i)3-s + (1.09 − 1.26i)4-s + (2.27 + 1.46i)5-s + (0.372 + 0.430i)6-s + (−0.481 − 3.35i)7-s + (−2.00 − 0.589i)8-s + (0.841 − 0.540i)9-s + (0.218 − 1.52i)10-s + (−1.86 + 4.08i)11-s + (−0.696 + 1.52i)12-s + (−0.891 + 6.19i)13-s + (−1.62 + 1.04i)14-s + (−2.59 − 0.761i)15-s + (−0.307 − 2.14i)16-s + (−2.00 − 2.31i)17-s + ⋯ |
| L(s) = 1 | + (−0.167 − 0.365i)2-s + (−0.553 + 0.162i)3-s + (0.548 − 0.633i)4-s + (1.01 + 0.653i)5-s + (0.152 + 0.175i)6-s + (−0.182 − 1.26i)7-s + (−0.709 − 0.208i)8-s + (0.280 − 0.180i)9-s + (0.0691 − 0.481i)10-s + (−0.562 + 1.23i)11-s + (−0.201 + 0.440i)12-s + (−0.247 + 1.71i)13-s + (−0.433 + 0.278i)14-s + (−0.669 − 0.196i)15-s + (−0.0769 − 0.535i)16-s + (−0.487 − 0.562i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 69 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.833 + 0.552i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 69 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.833 + 0.552i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.827233 - 0.249101i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.827233 - 0.249101i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (0.959 - 0.281i)T \) |
| 23 | \( 1 + (3.18 + 3.58i)T \) |
| good | 2 | \( 1 + (0.236 + 0.517i)T + (-1.30 + 1.51i)T^{2} \) |
| 5 | \( 1 + (-2.27 - 1.46i)T + (2.07 + 4.54i)T^{2} \) |
| 7 | \( 1 + (0.481 + 3.35i)T + (-6.71 + 1.97i)T^{2} \) |
| 11 | \( 1 + (1.86 - 4.08i)T + (-7.20 - 8.31i)T^{2} \) |
| 13 | \( 1 + (0.891 - 6.19i)T + (-12.4 - 3.66i)T^{2} \) |
| 17 | \( 1 + (2.00 + 2.31i)T + (-2.41 + 16.8i)T^{2} \) |
| 19 | \( 1 + (0.686 - 0.792i)T + (-2.70 - 18.8i)T^{2} \) |
| 29 | \( 1 + (-1.45 - 1.67i)T + (-4.12 + 28.7i)T^{2} \) |
| 31 | \( 1 + (-6.22 - 1.82i)T + (26.0 + 16.7i)T^{2} \) |
| 37 | \( 1 + (-0.235 + 0.151i)T + (15.3 - 33.6i)T^{2} \) |
| 41 | \( 1 + (-1.19 - 0.767i)T + (17.0 + 37.2i)T^{2} \) |
| 43 | \( 1 + (1.76 - 0.518i)T + (36.1 - 23.2i)T^{2} \) |
| 47 | \( 1 - 4.74T + 47T^{2} \) |
| 53 | \( 1 + (0.768 + 5.34i)T + (-50.8 + 14.9i)T^{2} \) |
| 59 | \( 1 + (-1.54 + 10.7i)T + (-56.6 - 16.6i)T^{2} \) |
| 61 | \( 1 + (4.49 + 1.31i)T + (51.3 + 32.9i)T^{2} \) |
| 67 | \( 1 + (0.0835 + 0.182i)T + (-43.8 + 50.6i)T^{2} \) |
| 71 | \( 1 + (1.25 + 2.74i)T + (-46.4 + 53.6i)T^{2} \) |
| 73 | \( 1 + (-7.91 + 9.13i)T + (-10.3 - 72.2i)T^{2} \) |
| 79 | \( 1 + (1.10 - 7.68i)T + (-75.7 - 22.2i)T^{2} \) |
| 83 | \( 1 + (12.4 - 7.98i)T + (34.4 - 75.4i)T^{2} \) |
| 89 | \( 1 + (-0.838 + 0.246i)T + (74.8 - 48.1i)T^{2} \) |
| 97 | \( 1 + (-9.73 - 6.25i)T + (40.2 + 88.2i)T^{2} \) |
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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
−14.47212870357085750542118621439, −13.76470239850506039430780420142, −12.23071758423173404830378295531, −11.02230742465949300047184137909, −10.15688752748739166576361989612, −9.645457574822689006687601009712, −7.00833471857656404436877683384, −6.45083936110494389611469100956, −4.62852841749353613180619164533, −2.11753764463620025196344524282,
2.68346020808410436689181169116, 5.61501160556929281047387706651, 6.01366034040810095718283150914, 7.965024324716745803416397984733, 8.876562185092268920422020515983, 10.38117450906931381172292531243, 11.71756581140546135016630424112, 12.70511870476942718175111200452, 13.41323463008140918764391685950, 15.35379158989706811909440527742
