| L(s) = 1 | + (−2.56 + 1.55i)3-s + (6.30 − 3.63i)5-s + (4.46 − 7.74i)7-s + (4.14 − 7.98i)9-s + (10.5 + 6.08i)11-s + (1.73 + 3.00i)13-s + (−10.4 + 19.1i)15-s − 33.3i·17-s − 33.3·19-s + (0.600 + 26.8i)21-s + (−15.8 + 9.16i)23-s + (13.9 − 24.2i)25-s + (1.81 + 26.9i)27-s + (−13.6 − 7.85i)29-s + (−6.66 − 11.5i)31-s + ⋯ |
| L(s) = 1 | + (−0.854 + 0.519i)3-s + (1.26 − 0.727i)5-s + (0.638 − 1.10i)7-s + (0.460 − 0.887i)9-s + (0.958 + 0.553i)11-s + (0.133 + 0.231i)13-s + (−0.699 + 1.27i)15-s − 1.95i·17-s − 1.75·19-s + (0.0285 + 1.27i)21-s + (−0.690 + 0.398i)23-s + (0.558 − 0.968i)25-s + (0.0671 + 0.997i)27-s + (−0.469 − 0.270i)29-s + (−0.215 − 0.372i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.383 + 0.923i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.383 + 0.923i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.716485598\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.716485598\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (2.56 - 1.55i)T \) |
| good | 5 | \( 1 + (-6.30 + 3.63i)T + (12.5 - 21.6i)T^{2} \) |
| 7 | \( 1 + (-4.46 + 7.74i)T + (-24.5 - 42.4i)T^{2} \) |
| 11 | \( 1 + (-10.5 - 6.08i)T + (60.5 + 104. i)T^{2} \) |
| 13 | \( 1 + (-1.73 - 3.00i)T + (-84.5 + 146. i)T^{2} \) |
| 17 | \( 1 + 33.3iT - 289T^{2} \) |
| 19 | \( 1 + 33.3T + 361T^{2} \) |
| 23 | \( 1 + (15.8 - 9.16i)T + (264.5 - 458. i)T^{2} \) |
| 29 | \( 1 + (13.6 + 7.85i)T + (420.5 + 728. i)T^{2} \) |
| 31 | \( 1 + (6.66 + 11.5i)T + (-480.5 + 832. i)T^{2} \) |
| 37 | \( 1 - 35.3T + 1.36e3T^{2} \) |
| 41 | \( 1 + (-4.55 + 2.62i)T + (840.5 - 1.45e3i)T^{2} \) |
| 43 | \( 1 + (-20.1 + 34.9i)T + (-924.5 - 1.60e3i)T^{2} \) |
| 47 | \( 1 + (-29.9 - 17.2i)T + (1.10e3 + 1.91e3i)T^{2} \) |
| 53 | \( 1 - 15.9iT - 2.80e3T^{2} \) |
| 59 | \( 1 + (-16.8 + 9.70i)T + (1.74e3 - 3.01e3i)T^{2} \) |
| 61 | \( 1 + (7.12 - 12.3i)T + (-1.86e3 - 3.22e3i)T^{2} \) |
| 67 | \( 1 + (16.8 + 29.1i)T + (-2.24e3 + 3.88e3i)T^{2} \) |
| 71 | \( 1 - 21.5iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 35.0T + 5.32e3T^{2} \) |
| 79 | \( 1 + (-75.5 + 130. i)T + (-3.12e3 - 5.40e3i)T^{2} \) |
| 83 | \( 1 + (107. + 62.2i)T + (3.44e3 + 5.96e3i)T^{2} \) |
| 89 | \( 1 - 58.9iT - 7.92e3T^{2} \) |
| 97 | \( 1 + (43.2 - 74.8i)T + (-4.70e3 - 8.14e3i)T^{2} \) |
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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
−10.30811719661015166381674041664, −9.517948273008669391899044046738, −9.024148374083126147306061655693, −7.48768534181259830007993241597, −6.56554757619281212788517674840, −5.68707197481507910719356028234, −4.64462304743509205510005630109, −4.12809121659930898953386556819, −1.94665006557285751480469728869, −0.75276868366518392702435074579,
1.61227747406290665394590723141, 2.32313447230037218344751198992, 4.15562549616945390718301740502, 5.62393399592159239127750186232, 6.09579088763986353423009537194, 6.61402094753426990667998737976, 8.139259752521999808335125368971, 8.818982054658806306590280785900, 10.05362620224058166006351544571, 10.79842591508141887412862804593
