L(s) = 1 | + (−1.62 − 1.16i)2-s + (−4.68 + 1.94i)3-s + (1.27 + 3.78i)4-s + (−4.51 − 1.86i)5-s + (9.87 + 2.31i)6-s + (−3.85 + 3.85i)7-s + (2.34 − 7.65i)8-s + (11.8 − 11.8i)9-s + (5.14 + 8.29i)10-s + (−4.56 − 1.89i)11-s + (−13.3 − 15.2i)12-s + (−5.58 + 2.31i)13-s + (10.7 − 1.76i)14-s + 24.7·15-s + (−12.7 + 9.70i)16-s + 25.0i·17-s + ⋯ |
L(s) = 1 | + (−0.812 − 0.583i)2-s + (−1.56 + 0.647i)3-s + (0.319 + 0.947i)4-s + (−0.902 − 0.373i)5-s + (1.64 + 0.385i)6-s + (−0.550 + 0.550i)7-s + (0.292 − 0.956i)8-s + (1.31 − 1.31i)9-s + (0.514 + 0.829i)10-s + (−0.414 − 0.171i)11-s + (−1.11 − 1.27i)12-s + (−0.429 + 0.177i)13-s + (0.768 − 0.126i)14-s + 1.65·15-s + (−0.795 + 0.606i)16-s + 1.47i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 32 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.847 - 0.531i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 32 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.847 - 0.531i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(0.0265890 + 0.0924223i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.0265890 + 0.0924223i\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (1.62 + 1.16i)T \) |
good | 3 | \( 1 + (4.68 - 1.94i)T + (6.36 - 6.36i)T^{2} \) |
| 5 | \( 1 + (4.51 + 1.86i)T + (17.6 + 17.6i)T^{2} \) |
| 7 | \( 1 + (3.85 - 3.85i)T - 49iT^{2} \) |
| 11 | \( 1 + (4.56 + 1.89i)T + (85.5 + 85.5i)T^{2} \) |
| 13 | \( 1 + (5.58 - 2.31i)T + (119. - 119. i)T^{2} \) |
| 17 | \( 1 - 25.0iT - 289T^{2} \) |
| 19 | \( 1 + (-6.43 - 15.5i)T + (-255. + 255. i)T^{2} \) |
| 23 | \( 1 + (26.9 + 26.9i)T + 529iT^{2} \) |
| 29 | \( 1 + (0.210 + 0.507i)T + (-594. + 594. i)T^{2} \) |
| 31 | \( 1 + 15.8iT - 961T^{2} \) |
| 37 | \( 1 + (2.18 + 0.905i)T + (968. + 968. i)T^{2} \) |
| 41 | \( 1 + (31.1 - 31.1i)T - 1.68e3iT^{2} \) |
| 43 | \( 1 + (-12.9 - 5.34i)T + (1.30e3 + 1.30e3i)T^{2} \) |
| 47 | \( 1 - 15.0T + 2.20e3T^{2} \) |
| 53 | \( 1 + (15.4 - 37.2i)T + (-1.98e3 - 1.98e3i)T^{2} \) |
| 59 | \( 1 + (14.7 - 35.5i)T + (-2.46e3 - 2.46e3i)T^{2} \) |
| 61 | \( 1 + (15.4 + 37.3i)T + (-2.63e3 + 2.63e3i)T^{2} \) |
| 67 | \( 1 + (-61.3 + 25.4i)T + (3.17e3 - 3.17e3i)T^{2} \) |
| 71 | \( 1 + (51.7 - 51.7i)T - 5.04e3iT^{2} \) |
| 73 | \( 1 + (-64.9 + 64.9i)T - 5.32e3iT^{2} \) |
| 79 | \( 1 + 38.1T + 6.24e3T^{2} \) |
| 83 | \( 1 + (-15.9 - 38.5i)T + (-4.87e3 + 4.87e3i)T^{2} \) |
| 89 | \( 1 + (-23.7 - 23.7i)T + 7.92e3iT^{2} \) |
| 97 | \( 1 + 118.T + 9.40e3T^{2} \) |
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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
−16.94088429818498239783085013889, −16.27828464318368103230785942523, −15.43944036301911306986813855671, −12.49810640749717450486286371887, −12.05929875179763094506645616308, −10.82301800542092900072673255385, −9.844604523252091589817594598515, −8.142993328345878188769520666111, −6.16122977039455051458623055301, −4.14288406269716082451246696331,
0.17693124480114067031269225390, 5.27723498960608772891195883872, 6.90558411161663554578036624369, 7.53234680993026455970787555760, 9.896615939571489765820980759729, 11.13207844218318692664463509653, 11.96636832530569192086634224194, 13.65588614386520164553360823419, 15.60149828352961656094268973054, 16.22570218090013295723565238917