| L(s) = 1 | + (0.194 + 0.725i)2-s + (−1.34 + 2.68i)3-s + (2.97 − 1.71i)4-s + (−1.25 + 4.83i)5-s + (−2.20 − 0.451i)6-s + (0.481 + 1.79i)7-s + (3.94 + 3.94i)8-s + (−5.40 − 7.19i)9-s + (−3.75 + 0.0302i)10-s + (5.82 − 10.0i)11-s + (0.617 + 10.2i)12-s + (5.30 − 19.8i)13-s + (−1.21 + 0.698i)14-s + (−11.3 − 9.86i)15-s + (4.77 − 8.26i)16-s + (−10.0 + 10.0i)17-s + ⋯ |
| L(s) = 1 | + (0.0972 + 0.362i)2-s + (−0.447 + 0.894i)3-s + (0.743 − 0.429i)4-s + (−0.251 + 0.967i)5-s + (−0.368 − 0.0753i)6-s + (0.0687 + 0.256i)7-s + (0.493 + 0.493i)8-s + (−0.600 − 0.799i)9-s + (−0.375 + 0.00302i)10-s + (0.529 − 0.916i)11-s + (0.0514 + 0.857i)12-s + (0.408 − 1.52i)13-s + (−0.0864 + 0.0499i)14-s + (−0.753 − 0.657i)15-s + (0.298 − 0.516i)16-s + (−0.589 + 0.589i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 45 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.331 - 0.943i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 45 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.331 - 0.943i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.935393 + 0.662462i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.935393 + 0.662462i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (1.34 - 2.68i)T \) |
| 5 | \( 1 + (1.25 - 4.83i)T \) |
| good | 2 | \( 1 + (-0.194 - 0.725i)T + (-3.46 + 2i)T^{2} \) |
| 7 | \( 1 + (-0.481 - 1.79i)T + (-42.4 + 24.5i)T^{2} \) |
| 11 | \( 1 + (-5.82 + 10.0i)T + (-60.5 - 104. i)T^{2} \) |
| 13 | \( 1 + (-5.30 + 19.8i)T + (-146. - 84.5i)T^{2} \) |
| 17 | \( 1 + (10.0 - 10.0i)T - 289iT^{2} \) |
| 19 | \( 1 - 10.8iT - 361T^{2} \) |
| 23 | \( 1 + (-0.360 + 1.34i)T + (-458. - 264.5i)T^{2} \) |
| 29 | \( 1 + (20.7 + 12.0i)T + (420.5 + 728. i)T^{2} \) |
| 31 | \( 1 + (-21.6 - 37.4i)T + (-480.5 + 832. i)T^{2} \) |
| 37 | \( 1 + (32.5 - 32.5i)T - 1.36e3iT^{2} \) |
| 41 | \( 1 + (20.5 + 35.5i)T + (-840.5 + 1.45e3i)T^{2} \) |
| 43 | \( 1 + (8.01 - 2.14i)T + (1.60e3 - 924.5i)T^{2} \) |
| 47 | \( 1 + (4.62 + 17.2i)T + (-1.91e3 + 1.10e3i)T^{2} \) |
| 53 | \( 1 + (51.3 + 51.3i)T + 2.80e3iT^{2} \) |
| 59 | \( 1 + (24.3 - 14.0i)T + (1.74e3 - 3.01e3i)T^{2} \) |
| 61 | \( 1 + (41.1 - 71.1i)T + (-1.86e3 - 3.22e3i)T^{2} \) |
| 67 | \( 1 + (32.3 + 8.65i)T + (3.88e3 + 2.24e3i)T^{2} \) |
| 71 | \( 1 - 99.6T + 5.04e3T^{2} \) |
| 73 | \( 1 + (22.3 + 22.3i)T + 5.32e3iT^{2} \) |
| 79 | \( 1 + (-52.9 - 30.5i)T + (3.12e3 + 5.40e3i)T^{2} \) |
| 83 | \( 1 + (49.9 - 13.3i)T + (5.96e3 - 3.44e3i)T^{2} \) |
| 89 | \( 1 - 113. iT - 7.92e3T^{2} \) |
| 97 | \( 1 + (-7.92 - 29.5i)T + (-8.14e3 + 4.70e3i)T^{2} \) |
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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
−15.51135516410617658352937177206, −15.15197066386859442081707471974, −13.98481698221463108634828022151, −11.91631194805264356566444046728, −10.88018667157208274654493153120, −10.28181504342028684829445155302, −8.328698075849074960436323594149, −6.54361766059756546525125134674, −5.58187185013802895393718611680, −3.35010191357311499804866784425,
1.77424650825442360593761824952, 4.44979219065437391890038186821, 6.57305454228139465296361086122, 7.58840911770947365647620433158, 9.172024225036839473534120758864, 11.19534870528576290796964908473, 11.87453793199498329341912954930, 12.79470578550207927095698756525, 13.82295500200595059096108032900, 15.66554638132357520451556938838
