L(s) = 1 | + (−190. + 190. i)3-s + (2.75e3 − 1.47e3i)5-s + (1.48e4 + 1.48e4i)7-s − 1.38e4i·9-s − 3.19e5·11-s + (−7.69e4 + 7.69e4i)13-s + (−2.44e5 + 8.07e5i)15-s + (−7.97e5 − 7.97e5i)17-s + 3.23e6i·19-s − 5.66e6·21-s + (−5.27e6 + 5.27e6i)23-s + (5.42e6 − 8.11e6i)25-s + (−8.63e6 − 8.63e6i)27-s + 2.58e7i·29-s + 1.45e7·31-s + ⋯ |
L(s) = 1 | + (−0.785 + 0.785i)3-s + (0.881 − 0.471i)5-s + (0.883 + 0.883i)7-s − 0.233i·9-s − 1.98·11-s + (−0.207 + 0.207i)13-s + (−0.322 + 1.06i)15-s + (−0.561 − 0.561i)17-s + 1.30i·19-s − 1.38·21-s + (−0.819 + 0.819i)23-s + (0.555 − 0.831i)25-s + (−0.601 − 0.601i)27-s + 1.25i·29-s + 0.508·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 20 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.864 - 0.502i)\, \overline{\Lambda}(11-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 20 ^{s/2} \, \Gamma_{\C}(s+5) \, L(s)\cr =\mathstrut & (-0.864 - 0.502i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{11}{2})\) |
\(\approx\) |
\(0.244889 + 0.908721i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.244889 + 0.908721i\) |
\(L(6)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + (-2.75e3 + 1.47e3i)T \) |
good | 3 | \( 1 + (190. - 190. i)T - 5.90e4iT^{2} \) |
| 7 | \( 1 + (-1.48e4 - 1.48e4i)T + 2.82e8iT^{2} \) |
| 11 | \( 1 + 3.19e5T + 2.59e10T^{2} \) |
| 13 | \( 1 + (7.69e4 - 7.69e4i)T - 1.37e11iT^{2} \) |
| 17 | \( 1 + (7.97e5 + 7.97e5i)T + 2.01e12iT^{2} \) |
| 19 | \( 1 - 3.23e6iT - 6.13e12T^{2} \) |
| 23 | \( 1 + (5.27e6 - 5.27e6i)T - 4.14e13iT^{2} \) |
| 29 | \( 1 - 2.58e7iT - 4.20e14T^{2} \) |
| 31 | \( 1 - 1.45e7T + 8.19e14T^{2} \) |
| 37 | \( 1 + (2.45e7 + 2.45e7i)T + 4.80e15iT^{2} \) |
| 41 | \( 1 + 6.80e7T + 1.34e16T^{2} \) |
| 43 | \( 1 + (5.22e6 - 5.22e6i)T - 2.16e16iT^{2} \) |
| 47 | \( 1 + (-4.11e7 - 4.11e7i)T + 5.25e16iT^{2} \) |
| 53 | \( 1 + (2.99e8 - 2.99e8i)T - 1.74e17iT^{2} \) |
| 59 | \( 1 + 4.68e7iT - 5.11e17T^{2} \) |
| 61 | \( 1 - 2.29e8T + 7.13e17T^{2} \) |
| 67 | \( 1 + (-1.40e9 - 1.40e9i)T + 1.82e18iT^{2} \) |
| 71 | \( 1 - 1.91e9T + 3.25e18T^{2} \) |
| 73 | \( 1 + (-2.58e9 + 2.58e9i)T - 4.29e18iT^{2} \) |
| 79 | \( 1 - 5.01e9iT - 9.46e18T^{2} \) |
| 83 | \( 1 + (-2.23e9 + 2.23e9i)T - 1.55e19iT^{2} \) |
| 89 | \( 1 - 7.96e9iT - 3.11e19T^{2} \) |
| 97 | \( 1 + (4.52e9 + 4.52e9i)T + 7.37e19iT^{2} \) |
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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
−16.37952507667391032939200304838, −15.49393354769510121580556863382, −13.86337601090972180245872693549, −12.36648755889847201153782319490, −10.91802044949414892123653055462, −9.831927219474265406094750433041, −8.177371246686362804235104961303, −5.58028559558017294842452669300, −4.98698903066272825885595275419, −2.12090337583122778846840120745,
0.41733292653824843150257176815, 2.21957458338221391986803129776, 5.07224218279528512065442912879, 6.55576650705049135492632154485, 7.895059124504254289687675532472, 10.26244167209522369150293630971, 11.17834438550338162280754347240, 12.91201371611757494969120369961, 13.75981559386201487692420295801, 15.36240464106389624025616074277