| L(s) = 1 | + (3.39 − 1.96i)5-s + (−6.39 + 11.0i)7-s + (14.2 + 8.25i)11-s + (1.39 + 2.42i)13-s − 2.54i·17-s + 21.5·19-s + (−2.60 + 1.50i)23-s + (−4.79 + 8.31i)25-s + (13.1 + 7.61i)29-s + (13.6 + 23.5i)31-s + 50.2i·35-s − 10.4·37-s + (34.5 − 19.9i)41-s + (−17.0 + 29.6i)43-s + (−58.1 − 33.6i)47-s + ⋯ |
| L(s) = 1 | + (0.679 − 0.392i)5-s + (−0.914 + 1.58i)7-s + (1.29 + 0.750i)11-s + (0.107 + 0.186i)13-s − 0.149i·17-s + 1.13·19-s + (−0.113 + 0.0652i)23-s + (−0.191 + 0.332i)25-s + (0.455 + 0.262i)29-s + (0.438 + 0.759i)31-s + 1.43i·35-s − 0.281·37-s + (0.841 − 0.485i)41-s + (−0.397 + 0.688i)43-s + (−1.23 − 0.714i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.642 - 0.766i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 216 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.642 - 0.766i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.46792 + 0.684505i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.46792 + 0.684505i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 + (-3.39 + 1.96i)T + (12.5 - 21.6i)T^{2} \) |
| 7 | \( 1 + (6.39 - 11.0i)T + (-24.5 - 42.4i)T^{2} \) |
| 11 | \( 1 + (-14.2 - 8.25i)T + (60.5 + 104. i)T^{2} \) |
| 13 | \( 1 + (-1.39 - 2.42i)T + (-84.5 + 146. i)T^{2} \) |
| 17 | \( 1 + 2.54iT - 289T^{2} \) |
| 19 | \( 1 - 21.5T + 361T^{2} \) |
| 23 | \( 1 + (2.60 - 1.50i)T + (264.5 - 458. i)T^{2} \) |
| 29 | \( 1 + (-13.1 - 7.61i)T + (420.5 + 728. i)T^{2} \) |
| 31 | \( 1 + (-13.6 - 23.5i)T + (-480.5 + 832. i)T^{2} \) |
| 37 | \( 1 + 10.4T + 1.36e3T^{2} \) |
| 41 | \( 1 + (-34.5 + 19.9i)T + (840.5 - 1.45e3i)T^{2} \) |
| 43 | \( 1 + (17.0 - 29.6i)T + (-924.5 - 1.60e3i)T^{2} \) |
| 47 | \( 1 + (58.1 + 33.6i)T + (1.10e3 + 1.91e3i)T^{2} \) |
| 53 | \( 1 + 100. iT - 2.80e3T^{2} \) |
| 59 | \( 1 + (5.29 - 3.05i)T + (1.74e3 - 3.01e3i)T^{2} \) |
| 61 | \( 1 + (20.3 - 35.3i)T + (-1.86e3 - 3.22e3i)T^{2} \) |
| 67 | \( 1 + (54.4 + 94.3i)T + (-2.24e3 + 3.88e3i)T^{2} \) |
| 71 | \( 1 - 52.8iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 68.7T + 5.32e3T^{2} \) |
| 79 | \( 1 + (-12.7 + 22.1i)T + (-3.12e3 - 5.40e3i)T^{2} \) |
| 83 | \( 1 + (52.0 + 30.0i)T + (3.44e3 + 5.96e3i)T^{2} \) |
| 89 | \( 1 + 7.62iT - 7.92e3T^{2} \) |
| 97 | \( 1 + (-50.7 + 87.8i)T + (-4.70e3 - 8.14e3i)T^{2} \) |
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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
−12.19539027645487816301785123278, −11.60251870798473024132220397400, −9.826648518381405052900806983574, −9.411961859858845872056746978977, −8.588548281022193822472016331604, −6.90655938291977709870789604798, −6.01602264883943329438804714068, −5.00995198927872568159379746450, −3.26921175411692204301333951809, −1.77643973216313871964612600047,
1.00169079006518252323017645352, 3.15919867959449680180291541181, 4.19252989300505559542051177619, 6.03562605672951683663904691059, 6.68589314274298376830796860914, 7.79486429105064905291758731884, 9.297630013063845249350199609793, 9.992806049203143722149579580137, 10.84018651317007093421321174787, 11.89145670723274549110722056217
