| L(s) = 1 | + (0.0492 + 0.183i)2-s + (−2.37 − 1.82i)3-s + (3.43 − 1.98i)4-s + (4.90 + 0.964i)5-s + (0.218 − 0.526i)6-s + (−2.33 − 8.70i)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 + (−11.7 − 1.56i)12-s + (−3.46 + 12.9i)13-s + (1.48 − 0.856i)14-s + (−9.90 − 11.2i)15-s + (7.78 − 13.4i)16-s + (−0.740 + 0.740i)17-s + ⋯ |
| L(s) = 1 | + (0.0246 + 0.0918i)2-s + (−0.792 − 0.609i)3-s + (0.858 − 0.495i)4-s + (0.981 + 0.192i)5-s + (0.0364 − 0.0877i)6-s + (−0.333 − 1.24i)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 + (−0.982 − 0.130i)12-s + (−0.266 + 0.995i)13-s + (0.105 − 0.0611i)14-s + (−0.660 − 0.751i)15-s + (0.486 − 0.842i)16-s + (−0.0435 + 0.0435i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 45 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.788 + 0.615i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 45 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.788 + 0.615i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.06164 - 0.365213i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.06164 - 0.365213i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (2.37 + 1.82i)T \) |
| 5 | \( 1 + (-4.90 - 0.964i)T \) |
| good | 2 | \( 1 + (-0.0492 - 0.183i)T + (-3.46 + 2i)T^{2} \) |
| 7 | \( 1 + (2.33 + 8.70i)T + (-42.4 + 24.5i)T^{2} \) |
| 11 | \( 1 + (7.04 - 12.1i)T + (-60.5 - 104. i)T^{2} \) |
| 13 | \( 1 + (3.46 - 12.9i)T + (-146. - 84.5i)T^{2} \) |
| 17 | \( 1 + (0.740 - 0.740i)T - 289iT^{2} \) |
| 19 | \( 1 + 7.09iT - 361T^{2} \) |
| 23 | \( 1 + (5.10 - 19.0i)T + (-458. - 264.5i)T^{2} \) |
| 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 | \( 1 + (12.0 - 3.22i)T + (1.60e3 - 924.5i)T^{2} \) |
| 47 | \( 1 + (13.9 + 51.8i)T + (-1.91e3 + 1.10e3i)T^{2} \) |
| 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 | \( 1 + (-64.1 - 17.1i)T + (3.88e3 + 2.24e3i)T^{2} \) |
| 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 | \( 1 + (-51.3 + 13.7i)T + (5.96e3 - 3.44e3i)T^{2} \) |
| 89 | \( 1 + 22.5iT - 7.92e3T^{2} \) |
| 97 | \( 1 + (9.03 + 33.7i)T + (-8.14e3 + 4.70e3i)T^{2} \) |
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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
−15.65055136046647046713109952597, −14.12920482446677572920691979177, −13.26652378479938301332814206953, −11.90146380532856324075556681799, −10.59316336174269487270584544959, −9.946888474611894721765627637532, −7.22984870917442444769209547212, −6.68886094895842742682931707378, −5.14491951726311954575047487350, −1.86147855442524058216581738880,
2.86578438158306193052860859624, 5.48149233137073482344924466037, 6.27568119541693274652195504756, 8.435484229737952826793837026219, 9.927289043958000650984569499952, 10.97491857996281393942867960954, 12.19081373603909683436400386228, 13.04442193312591905189723255184, 14.94273433101468109765290995636, 15.94345871711791226015778432804
