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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.766 + 0.642i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.766 + 0.642i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(2.165752324\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.165752324\) |
\(L(2)\) |
|
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
−9.201719819119637423714588258085, −7.897960870071367428210692524954, −7.45544709034305669784940084268, −7.01883760290639539740808518151, −5.85342773715107672008434912397, −4.58373093035433455160111871054, −4.13439987746887427293256284115, −3.28898008556305406750906962997, −1.70522542282908230202238905839, −0.73881280860320683689547438940,
1.01240289013010444192437792875, 2.10877082959110341609685153820, 3.32676624690273482802475625831, 4.23597954655924250744823218999, 5.25754811177061994201952340687, 5.79317661066175031993190620464, 6.85768134978899043135369087148, 7.82212234694627130870562507234, 8.650635283850914287842021961278, 8.897289024131762312160103666795