| 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
−15.66554638132357520451556938838, −13.82295500200595059096108032900, −12.79470578550207927095698756525, −11.87453793199498329341912954930, −11.19534870528576290796964908473, −9.172024225036839473534120758864, −7.58840911770947365647620433158, −6.57305454228139465296361086122, −4.44979219065437391890038186821, −1.77424650825442360593761824952,
3.35010191357311499804866784425, 5.58187185013802895393718611680, 6.54361766059756546525125134674, 8.328698075849074960436323594149, 10.28181504342028684829445155302, 10.88018667157208274654493153120, 11.91631194805264356566444046728, 13.98481698221463108634828022151, 15.15197066386859442081707471974, 15.51135516410617658352937177206
