| L(s) = 1 | + (−0.753 − 0.274i)2-s + (−1.03 − 0.872i)4-s + (0.477 − 2.70i)5-s + (1.82 − 1.52i)7-s + (1.34 + 2.33i)8-s + (−1.10 + 1.90i)10-s + (0.0434 + 0.246i)11-s + (−2.45 + 0.893i)13-s + (−1.79 + 0.651i)14-s + (0.0969 + 0.549i)16-s + (−0.146 + 0.254i)17-s + (1.39 + 2.41i)19-s + (−2.85 + 2.39i)20-s + (0.0348 − 0.197i)22-s + (5.12 + 4.30i)23-s + ⋯ |
| L(s) = 1 | + (−0.532 − 0.193i)2-s + (−0.519 − 0.436i)4-s + (0.213 − 1.21i)5-s + (0.688 − 0.577i)7-s + (0.475 + 0.823i)8-s + (−0.348 + 0.603i)10-s + (0.0130 + 0.0742i)11-s + (−0.680 + 0.247i)13-s + (−0.478 + 0.174i)14-s + (0.0242 + 0.137i)16-s + (−0.0355 + 0.0616i)17-s + (0.319 + 0.553i)19-s + (−0.639 + 0.536i)20-s + (0.00742 − 0.0420i)22-s + (1.06 + 0.896i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 81 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.301 + 0.953i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 81 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.301 + 0.953i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.573690 - 0.420206i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.573690 - 0.420206i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| good | 2 | \( 1 + (0.753 + 0.274i)T + (1.53 + 1.28i)T^{2} \) |
| 5 | \( 1 + (-0.477 + 2.70i)T + (-4.69 - 1.71i)T^{2} \) |
| 7 | \( 1 + (-1.82 + 1.52i)T + (1.21 - 6.89i)T^{2} \) |
| 11 | \( 1 + (-0.0434 - 0.246i)T + (-10.3 + 3.76i)T^{2} \) |
| 13 | \( 1 + (2.45 - 0.893i)T + (9.95 - 8.35i)T^{2} \) |
| 17 | \( 1 + (0.146 - 0.254i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-1.39 - 2.41i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-5.12 - 4.30i)T + (3.99 + 22.6i)T^{2} \) |
| 29 | \( 1 + (0.333 + 0.121i)T + (22.2 + 18.6i)T^{2} \) |
| 31 | \( 1 + (-2.11 - 1.77i)T + (5.38 + 30.5i)T^{2} \) |
| 37 | \( 1 + (-3.49 + 6.05i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (9.13 - 3.32i)T + (31.4 - 26.3i)T^{2} \) |
| 43 | \( 1 + (-0.0452 - 0.256i)T + (-40.4 + 14.7i)T^{2} \) |
| 47 | \( 1 + (-8.75 + 7.34i)T + (8.16 - 46.2i)T^{2} \) |
| 53 | \( 1 + 5.43T + 53T^{2} \) |
| 59 | \( 1 + (1.03 - 5.88i)T + (-55.4 - 20.1i)T^{2} \) |
| 61 | \( 1 + (9.07 - 7.61i)T + (10.5 - 60.0i)T^{2} \) |
| 67 | \( 1 + (1.70 - 0.619i)T + (51.3 - 43.0i)T^{2} \) |
| 71 | \( 1 + (0.185 - 0.320i)T + (-35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + (2.51 + 4.35i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (0.754 + 0.274i)T + (60.5 + 50.7i)T^{2} \) |
| 83 | \( 1 + (2.58 + 0.942i)T + (63.5 + 53.3i)T^{2} \) |
| 89 | \( 1 + (-5.22 - 9.05i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (2.57 + 14.6i)T + (-91.1 + 33.1i)T^{2} \) |
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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
−14.03115487621755002186978317361, −13.18884665265709585647990430793, −11.92796588240086392003422829441, −10.66900091149249135200831714070, −9.561891604673528697294728324869, −8.720036122648035933283018492900, −7.55913224959694942058308933546, −5.41300464380826649732324623648, −4.50418679033153046507861486613, −1.36238949171048255313363429519,
2.89633575768505488749041643482, 4.85924815146617231226069224434, 6.68265611651966187175592938912, 7.76825027649911468376054071092, 8.925964057048762524095118223947, 10.07852245816748326358202970412, 11.16858405132848361462894243814, 12.40442537623368912051654595669, 13.65897207663166899500914290580, 14.64571727848822117510187061342
