| 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
−15.35379158989706811909440527742, −13.41323463008140918764391685950, −12.70511870476942718175111200452, −11.71756581140546135016630424112, −10.38117450906931381172292531243, −8.876562185092268920422020515983, −7.965024324716745803416397984733, −6.01366034040810095718283150914, −5.61501160556929281047387706651, −2.68346020808410436689181169116,
2.11753764463620025196344524282, 4.62852841749353613180619164533, 6.45083936110494389611469100956, 7.00833471857656404436877683384, 9.645457574822689006687601009712, 10.15688752748739166576361989612, 11.02230742465949300047184137909, 12.23071758423173404830378295531, 13.76470239850506039430780420142, 14.47212870357085750542118621439
