| 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
−14.64571727848822117510187061342, −13.65897207663166899500914290580, −12.40442537623368912051654595669, −11.16858405132848361462894243814, −10.07852245816748326358202970412, −8.925964057048762524095118223947, −7.76825027649911468376054071092, −6.68265611651966187175592938912, −4.85924815146617231226069224434, −2.89633575768505488749041643482,
1.36238949171048255313363429519, 4.50418679033153046507861486613, 5.41300464380826649732324623648, 7.55913224959694942058308933546, 8.720036122648035933283018492900, 9.561891604673528697294728324869, 10.66900091149249135200831714070, 11.92796588240086392003422829441, 13.18884665265709585647990430793, 14.03115487621755002186978317361
