| L(s) = 1 | + (−1.65 − 2.50i)3-s + (0.721 − 0.416i)5-s + (1.26 − 2.18i)7-s + (−3.53 + 8.27i)9-s + (−9.47 − 5.47i)11-s + (−4.36 − 7.55i)13-s + (−2.23 − 1.11i)15-s + 20.8i·17-s − 1.50·19-s + (−7.54 + 0.454i)21-s + (−1.00 + 0.578i)23-s + (−12.1 + 21.0i)25-s + (26.5 − 4.84i)27-s + (−15.7 − 9.08i)29-s + (−25.6 − 44.4i)31-s + ⋯ |
| L(s) = 1 | + (−0.551 − 0.834i)3-s + (0.144 − 0.0832i)5-s + (0.180 − 0.311i)7-s + (−0.392 + 0.919i)9-s + (−0.861 − 0.497i)11-s + (−0.335 − 0.581i)13-s + (−0.148 − 0.0744i)15-s + 1.22i·17-s − 0.0790·19-s + (−0.359 + 0.0216i)21-s + (−0.0435 + 0.0251i)23-s + (−0.486 + 0.842i)25-s + (0.983 − 0.179i)27-s + (−0.542 − 0.313i)29-s + (−0.828 − 1.43i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.452 - 0.891i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.452 - 0.891i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.1420411536\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1420411536\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (1.65 + 2.50i)T \) |
| good | 5 | \( 1 + (-0.721 + 0.416i)T + (12.5 - 21.6i)T^{2} \) |
| 7 | \( 1 + (-1.26 + 2.18i)T + (-24.5 - 42.4i)T^{2} \) |
| 11 | \( 1 + (9.47 + 5.47i)T + (60.5 + 104. i)T^{2} \) |
| 13 | \( 1 + (4.36 + 7.55i)T + (-84.5 + 146. i)T^{2} \) |
| 17 | \( 1 - 20.8iT - 289T^{2} \) |
| 19 | \( 1 + 1.50T + 361T^{2} \) |
| 23 | \( 1 + (1.00 - 0.578i)T + (264.5 - 458. i)T^{2} \) |
| 29 | \( 1 + (15.7 + 9.08i)T + (420.5 + 728. i)T^{2} \) |
| 31 | \( 1 + (25.6 + 44.4i)T + (-480.5 + 832. i)T^{2} \) |
| 37 | \( 1 - 7.93T + 1.36e3T^{2} \) |
| 41 | \( 1 + (21.8 - 12.6i)T + (840.5 - 1.45e3i)T^{2} \) |
| 43 | \( 1 + (19.3 - 33.5i)T + (-924.5 - 1.60e3i)T^{2} \) |
| 47 | \( 1 + (-59.6 - 34.4i)T + (1.10e3 + 1.91e3i)T^{2} \) |
| 53 | \( 1 - 46.5iT - 2.80e3T^{2} \) |
| 59 | \( 1 + (89.1 - 51.4i)T + (1.74e3 - 3.01e3i)T^{2} \) |
| 61 | \( 1 + (44.1 - 76.4i)T + (-1.86e3 - 3.22e3i)T^{2} \) |
| 67 | \( 1 + (-11.3 - 19.6i)T + (-2.24e3 + 3.88e3i)T^{2} \) |
| 71 | \( 1 - 104. iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 75.2T + 5.32e3T^{2} \) |
| 79 | \( 1 + (-51.8 + 89.8i)T + (-3.12e3 - 5.40e3i)T^{2} \) |
| 83 | \( 1 + (53.7 + 31.0i)T + (3.44e3 + 5.96e3i)T^{2} \) |
| 89 | \( 1 - 1.95iT - 7.92e3T^{2} \) |
| 97 | \( 1 + (-59.2 + 102. i)T + (-4.70e3 - 8.14e3i)T^{2} \) |
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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
−10.85733369222221715752944393192, −10.19504846852235187081344063074, −8.920179527688782670562715014204, −7.78883779330419789963098271385, −7.50005355098402734638213927746, −6.00325392531820015225026556869, −5.63563136076709397318296211579, −4.29852454790744554785824223446, −2.76648720993658081120129798856, −1.45674825101403057450814225315,
0.05671497052860884516125591841, 2.20556674324524352417737642225, 3.53471684823887757337437459670, 4.84517637532556337948418045441, 5.30012726604039642451968128257, 6.53670025532777651554678777260, 7.45160377047705106175440696297, 8.719963498606355510801968166803, 9.474803956299502163672271152381, 10.26668110094478457831271369784
