L(s) = 1 | − 5.01i·3-s + (−46.1 − 31.5i)5-s + 15.3i·7-s + 217.·9-s + 576.·11-s + 607. i·13-s + (−158. + 231. i)15-s − 2.01e3i·17-s − 2.42e3·19-s + 77.2·21-s − 4.35e3i·23-s + (1.13e3 + 2.91e3i)25-s − 2.31e3i·27-s − 1.22e3·29-s − 7.81e3·31-s + ⋯ |
L(s) = 1 | − 0.321i·3-s + (−0.825 − 0.564i)5-s + 0.118i·7-s + 0.896·9-s + 1.43·11-s + 0.996i·13-s + (−0.181 + 0.265i)15-s − 1.68i·17-s − 1.53·19-s + 0.0382·21-s − 1.71i·23-s + (0.362 + 0.931i)25-s − 0.610i·27-s − 0.270·29-s − 1.46·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.564 + 0.825i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 160 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.564 + 0.825i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(3)\) |
\(\approx\) |
\(1.283816693\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.283816693\) |
\(L(\frac{7}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + (46.1 + 31.5i)T \) |
good | 3 | \( 1 + 5.01iT - 243T^{2} \) |
| 7 | \( 1 - 15.3iT - 1.68e4T^{2} \) |
| 11 | \( 1 - 576.T + 1.61e5T^{2} \) |
| 13 | \( 1 - 607. iT - 3.71e5T^{2} \) |
| 17 | \( 1 + 2.01e3iT - 1.41e6T^{2} \) |
| 19 | \( 1 + 2.42e3T + 2.47e6T^{2} \) |
| 23 | \( 1 + 4.35e3iT - 6.43e6T^{2} \) |
| 29 | \( 1 + 1.22e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 7.81e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 3.80e3iT - 6.93e7T^{2} \) |
| 41 | \( 1 + 4.47e3T + 1.15e8T^{2} \) |
| 43 | \( 1 + 1.47e4iT - 1.47e8T^{2} \) |
| 47 | \( 1 + 2.04e4iT - 2.29e8T^{2} \) |
| 53 | \( 1 - 1.09e4iT - 4.18e8T^{2} \) |
| 59 | \( 1 + 3.11e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 4.48e3T + 8.44e8T^{2} \) |
| 67 | \( 1 + 4.54e4iT - 1.35e9T^{2} \) |
| 71 | \( 1 - 1.03e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 3.89e4iT - 2.07e9T^{2} \) |
| 79 | \( 1 + 2.32e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 3.47e4iT - 3.93e9T^{2} \) |
| 89 | \( 1 - 9.78e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 1.51e5iT - 8.58e9T^{2} \) |
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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
−11.94221681345913474257371748544, −10.79501578585075998714675500527, −9.307323708137594818622729184825, −8.685588803878599103263311598054, −7.24197785944668967290405716665, −6.58383714632267321783030431326, −4.70193383918755553994475902284, −3.92440627330202637822561383737, −1.88971198984913344762289328174, −0.44340113818074821204598398380,
1.51679906694415746404612464742, 3.59763246624247033366688291669, 4.18770628446458297905612463692, 6.00535022649768962148828450780, 7.10759917343061517852431172942, 8.116000426110748597562191185616, 9.330329510558209817395931988758, 10.50373804631959298937961347021, 11.13198770633113980624604897084, 12.37702838192472968805985796765