| L(s) = 1 | + (−1.03 − 2.39i)2-s + (−3.29 + 3.49i)4-s + (0.0188 − 0.323i)5-s + (−3.75 + 0.889i)7-s + (6.85 + 2.49i)8-s + (−0.794 + 0.289i)10-s + (1.05 − 0.692i)11-s + (3.90 − 0.456i)13-s + (6.00 + 8.06i)14-s + (−0.548 − 9.41i)16-s + (3.68 + 3.09i)17-s + (−3.47 + 2.91i)19-s + (1.06 + 1.13i)20-s + (−2.74 − 1.80i)22-s + (0.546 + 0.129i)23-s + ⋯ |
| L(s) = 1 | + (−0.730 − 1.69i)2-s + (−1.64 + 1.74i)4-s + (0.00842 − 0.144i)5-s + (−1.41 + 0.336i)7-s + (2.42 + 0.882i)8-s + (−0.251 + 0.0914i)10-s + (0.317 − 0.208i)11-s + (1.08 − 0.126i)13-s + (1.60 + 2.15i)14-s + (−0.137 − 2.35i)16-s + (0.894 + 0.750i)17-s + (−0.798 + 0.669i)19-s + (0.238 + 0.252i)20-s + (−0.584 − 0.384i)22-s + (0.113 + 0.0270i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.121 + 0.992i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.121 + 0.992i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.524343 - 0.592314i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.524343 - 0.592314i\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + (1.03 + 2.39i)T + (-1.37 + 1.45i)T^{2} \) |
| 5 | \( 1 + (-0.0188 + 0.323i)T + (-4.96 - 0.580i)T^{2} \) |
| 7 | \( 1 + (3.75 - 0.889i)T + (6.25 - 3.14i)T^{2} \) |
| 11 | \( 1 + (-1.05 + 0.692i)T + (4.35 - 10.1i)T^{2} \) |
| 13 | \( 1 + (-3.90 + 0.456i)T + (12.6 - 2.99i)T^{2} \) |
| 17 | \( 1 + (-3.68 - 3.09i)T + (2.95 + 16.7i)T^{2} \) |
| 19 | \( 1 + (3.47 - 2.91i)T + (3.29 - 18.7i)T^{2} \) |
| 23 | \( 1 + (-0.546 - 0.129i)T + (20.5 + 10.3i)T^{2} \) |
| 29 | \( 1 + (-1.27 + 1.71i)T + (-8.31 - 27.7i)T^{2} \) |
| 31 | \( 1 + (1.09 + 3.64i)T + (-25.9 + 17.0i)T^{2} \) |
| 37 | \( 1 + (0.248 - 1.40i)T + (-34.7 - 12.6i)T^{2} \) |
| 41 | \( 1 + (-3.74 + 8.67i)T + (-28.1 - 29.8i)T^{2} \) |
| 43 | \( 1 + (3.23 + 1.62i)T + (25.6 + 34.4i)T^{2} \) |
| 47 | \( 1 + (-1.09 + 3.64i)T + (-39.2 - 25.8i)T^{2} \) |
| 53 | \( 1 + (-3.57 + 6.18i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (-4.88 - 3.21i)T + (23.3 + 54.1i)T^{2} \) |
| 61 | \( 1 + (-6.33 - 6.71i)T + (-3.54 + 60.8i)T^{2} \) |
| 67 | \( 1 + (-0.128 - 0.173i)T + (-19.2 + 64.1i)T^{2} \) |
| 71 | \( 1 + (-11.2 + 4.07i)T + (54.3 - 45.6i)T^{2} \) |
| 73 | \( 1 + (-7.37 - 2.68i)T + (55.9 + 46.9i)T^{2} \) |
| 79 | \( 1 + (-3.30 - 7.65i)T + (-54.2 + 57.4i)T^{2} \) |
| 83 | \( 1 + (0.262 + 0.608i)T + (-56.9 + 60.3i)T^{2} \) |
| 89 | \( 1 + (1.56 + 0.571i)T + (68.1 + 57.2i)T^{2} \) |
| 97 | \( 1 + (-0.937 - 16.0i)T + (-96.3 + 11.2i)T^{2} \) |
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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
−10.25669298999836638444840869917, −9.494183802663948769528647030069, −8.740534040766104795557340452460, −8.150849386864711554381077326876, −6.67340605395283563506778200568, −5.63828060478807037255595275577, −3.88847699231860749875559143694, −3.48248596121123786594946011899, −2.26897117437075100950727229762, −0.859407275539178123671706655762,
0.826525632047572026667549299052, 3.24818898054346919950674196301, 4.55054142414727229154061949051, 5.67781615915916508480536291787, 6.69095845062235646827590394039, 6.78209082738240200589687516993, 7.960992527753531860140742373485, 8.869751961979602523334198933218, 9.466806063019713531166101107500, 10.19516920609599356596645687613
