L(s) = 1 | − 3·3-s + (−3.39 − 1.96i)5-s + (−6.39 − 11.0i)7-s + 9·9-s + (−14.2 + 8.25i)11-s + (1.39 − 2.42i)13-s + (10.1 + 5.88i)15-s − 2.54i·17-s + 21.5·19-s + (19.1 + 33.2i)21-s + (2.60 + 1.50i)23-s + (−4.79 − 8.31i)25-s − 27·27-s + (−13.1 + 7.61i)29-s + (13.6 − 23.5i)31-s + ⋯ |
L(s) = 1 | − 3-s + (−0.679 − 0.392i)5-s + (−0.914 − 1.58i)7-s + 9-s + (−1.29 + 0.750i)11-s + (0.107 − 0.186i)13-s + (0.679 + 0.392i)15-s − 0.149i·17-s + 1.13·19-s + (0.914 + 1.58i)21-s + (0.113 + 0.0652i)23-s + (−0.191 − 0.332i)25-s − 27-s + (−0.455 + 0.262i)29-s + (0.438 − 0.759i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 72 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.766 + 0.642i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 72 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.766 + 0.642i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(0.138636 - 0.380900i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.138636 - 0.380900i\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + 3T \) |
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
−13.59004308823334235528872461767, −12.86930789109773101982957259921, −11.79099677019384335963492907074, −10.51955672629733634071606113790, −9.874256955944495777324945724807, −7.72122916025946336034105256645, −6.93641304668574602790362058676, −5.19515599657816200676076990932, −3.87612702068388476925627224306, −0.38875219257853939907942639391,
3.09948825449836774647032055976, 5.27117917312238473027574597001, 6.22772450729209262333574000741, 7.70966792063789311986459036415, 9.235921249891579579258880486514, 10.55446849568289845043810399217, 11.62976829804033647202821160670, 12.38191704306004961101355924819, 13.46506825282702722313382925801, 15.40843045297638279433707783263