| L(s) = 1 | + (−0.183 + 0.0492i)2-s + (−1.82 + 2.37i)3-s + (−3.43 + 1.98i)4-s + (−1.61 + 4.73i)5-s + (0.218 − 0.526i)6-s + (8.70 − 2.33i)7-s + (1.07 − 1.07i)8-s + (−2.30 − 8.69i)9-s + (0.0643 − 0.948i)10-s + (−7.04 + 12.1i)11-s + (1.56 − 11.7i)12-s + (12.9 + 3.46i)13-s + (−1.48 + 0.856i)14-s + (−8.29 − 12.5i)15-s + (7.78 − 13.4i)16-s + (−0.740 − 0.740i)17-s + ⋯ |
| L(s) = 1 | + (−0.0918 + 0.0246i)2-s + (−0.609 + 0.792i)3-s + (−0.858 + 0.495i)4-s + (−0.323 + 0.946i)5-s + (0.0364 − 0.0877i)6-s + (1.24 − 0.333i)7-s + (0.133 − 0.133i)8-s + (−0.256 − 0.966i)9-s + (0.00643 − 0.0948i)10-s + (−0.640 + 1.10i)11-s + (0.130 − 0.982i)12-s + (0.995 + 0.266i)13-s + (−0.105 + 0.0611i)14-s + (−0.552 − 0.833i)15-s + (0.486 − 0.842i)16-s + (−0.0435 − 0.0435i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 45 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.304 - 0.952i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 45 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.304 - 0.952i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.438024 + 0.599624i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.438024 + 0.599624i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (1.82 - 2.37i)T \) |
| 5 | \( 1 + (1.61 - 4.73i)T \) |
| good | 2 | \( 1 + (0.183 - 0.0492i)T + (3.46 - 2i)T^{2} \) |
| 7 | \( 1 + (-8.70 + 2.33i)T + (42.4 - 24.5i)T^{2} \) |
| 11 | \( 1 + (7.04 - 12.1i)T + (-60.5 - 104. i)T^{2} \) |
| 13 | \( 1 + (-12.9 - 3.46i)T + (146. + 84.5i)T^{2} \) |
| 17 | \( 1 + (0.740 + 0.740i)T + 289iT^{2} \) |
| 19 | \( 1 - 7.09iT - 361T^{2} \) |
| 23 | \( 1 + (-19.0 - 5.10i)T + (458. + 264.5i)T^{2} \) |
| 29 | \( 1 + (-6.18 - 3.56i)T + (420.5 + 728. i)T^{2} \) |
| 31 | \( 1 + (-13.0 - 22.6i)T + (-480.5 + 832. i)T^{2} \) |
| 37 | \( 1 + (23.0 + 23.0i)T + 1.36e3iT^{2} \) |
| 41 | \( 1 + (36.0 + 62.4i)T + (-840.5 + 1.45e3i)T^{2} \) |
| 43 | \( 1 + (-3.22 - 12.0i)T + (-1.60e3 + 924.5i)T^{2} \) |
| 47 | \( 1 + (-51.8 + 13.9i)T + (1.91e3 - 1.10e3i)T^{2} \) |
| 53 | \( 1 + (-17.2 + 17.2i)T - 2.80e3iT^{2} \) |
| 59 | \( 1 + (-27.5 + 15.9i)T + (1.74e3 - 3.01e3i)T^{2} \) |
| 61 | \( 1 + (-40.7 + 70.5i)T + (-1.86e3 - 3.22e3i)T^{2} \) |
| 67 | \( 1 + (17.1 - 64.1i)T + (-3.88e3 - 2.24e3i)T^{2} \) |
| 71 | \( 1 + 36.3T + 5.04e3T^{2} \) |
| 73 | \( 1 + (-2.90 + 2.90i)T - 5.32e3iT^{2} \) |
| 79 | \( 1 + (-71.8 - 41.5i)T + (3.12e3 + 5.40e3i)T^{2} \) |
| 83 | \( 1 + (13.7 + 51.3i)T + (-5.96e3 + 3.44e3i)T^{2} \) |
| 89 | \( 1 - 22.5iT - 7.92e3T^{2} \) |
| 97 | \( 1 + (-33.7 + 9.03i)T + (8.14e3 - 4.70e3i)T^{2} \) |
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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
−15.89438487291985890879125042611, −14.85890945805189921885544123409, −13.88236055448258918795121965180, −12.23592052988265342103334228210, −11.05736297678091274984209792909, −10.15888185448581514220031954744, −8.592676222475213764903150809414, −7.20694133080755174338001261795, −5.08047746031290113625727923247, −3.85179218546477605945671407091,
1.02363506863875025567228697477, 4.81979266306017987444491851311, 5.76932295554905877743543043240, 8.100902379055240278230184793114, 8.676000757629839711610931193695, 10.75514440625857173391516560123, 11.68728203781221840818739866094, 13.13282574367354337053011950237, 13.72449204180259284987722676076, 15.28182927739155403611467380784
