| L(s) = 1 | + (0.485 + 1.66i)3-s + (−0.582 − 0.489i)5-s + (3.39 + 1.23i)7-s + (−2.52 + 1.61i)9-s + (1.22 − 1.02i)11-s + (−0.872 + 4.94i)13-s + (0.530 − 1.20i)15-s + (−0.153 + 0.265i)17-s + (−0.463 − 0.802i)19-s + (−0.405 + 6.24i)21-s + (−1.41 + 0.514i)23-s + (−0.767 − 4.35i)25-s + (−3.91 − 3.41i)27-s + (−0.935 − 5.30i)29-s + (8.64 − 3.14i)31-s + ⋯ |
| L(s) = 1 | + (0.280 + 0.959i)3-s + (−0.260 − 0.218i)5-s + (1.28 + 0.466i)7-s + (−0.842 + 0.538i)9-s + (0.368 − 0.309i)11-s + (−0.241 + 1.37i)13-s + (0.136 − 0.311i)15-s + (−0.0371 + 0.0643i)17-s + (−0.106 − 0.184i)19-s + (−0.0884 + 1.36i)21-s + (−0.294 + 0.107i)23-s + (−0.153 − 0.870i)25-s + (−0.753 − 0.657i)27-s + (−0.173 − 0.984i)29-s + (1.55 − 0.565i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.466 - 0.884i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 216 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.466 - 0.884i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.16749 + 0.704153i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.16749 + 0.704153i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (-0.485 - 1.66i)T \) |
| good | 5 | \( 1 + (0.582 + 0.489i)T + (0.868 + 4.92i)T^{2} \) |
| 7 | \( 1 + (-3.39 - 1.23i)T + (5.36 + 4.49i)T^{2} \) |
| 11 | \( 1 + (-1.22 + 1.02i)T + (1.91 - 10.8i)T^{2} \) |
| 13 | \( 1 + (0.872 - 4.94i)T + (-12.2 - 4.44i)T^{2} \) |
| 17 | \( 1 + (0.153 - 0.265i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (0.463 + 0.802i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (1.41 - 0.514i)T + (17.6 - 14.7i)T^{2} \) |
| 29 | \( 1 + (0.935 + 5.30i)T + (-27.2 + 9.91i)T^{2} \) |
| 31 | \( 1 + (-8.64 + 3.14i)T + (23.7 - 19.9i)T^{2} \) |
| 37 | \( 1 + (3.04 - 5.27i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (-2.06 + 11.7i)T + (-38.5 - 14.0i)T^{2} \) |
| 43 | \( 1 + (-5.66 + 4.74i)T + (7.46 - 42.3i)T^{2} \) |
| 47 | \( 1 + (-7.10 - 2.58i)T + (36.0 + 30.2i)T^{2} \) |
| 53 | \( 1 + 10.5T + 53T^{2} \) |
| 59 | \( 1 + (9.36 + 7.85i)T + (10.2 + 58.1i)T^{2} \) |
| 61 | \( 1 + (5.59 + 2.03i)T + (46.7 + 39.2i)T^{2} \) |
| 67 | \( 1 + (0.432 - 2.45i)T + (-62.9 - 22.9i)T^{2} \) |
| 71 | \( 1 + (5.68 - 9.84i)T + (-35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + (4.80 + 8.32i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (0.450 + 2.55i)T + (-74.2 + 27.0i)T^{2} \) |
| 83 | \( 1 + (-2.82 - 16.0i)T + (-77.9 + 28.3i)T^{2} \) |
| 89 | \( 1 + (-4.62 - 8.01i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (-2.67 + 2.24i)T + (16.8 - 95.5i)T^{2} \) |
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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
−12.05348984162794469921156228647, −11.57590266900958952778262607797, −10.61090471067584624337569244488, −9.408726910652590686062241117893, −8.643131263558140412397206382997, −7.80467775532748712518173449524, −6.12920564085428199835114162126, −4.79482895910994484867260773450, −4.10141554340883006343268704189, −2.24375310554850262360626089138,
1.38465979288485926636550311135, 3.06635654755071105216156235479, 4.70911855257954493318870980035, 6.05952005116246908759034171048, 7.44650663606053899460767446360, 7.84571777489301518667641014608, 8.938435929989398248654396132990, 10.41509289898726922097809610997, 11.29875752777219623471569547158, 12.19962472351593950460090911013
