L(s) = 1 | + (−2.34 − 1.35i)2-s + (2.66 + 4.62i)4-s + 1.20·5-s − 9.04i·8-s + (−2.82 − 1.62i)10-s + 2.48i·11-s + (1.63 + 0.942i)13-s + (−6.90 + 11.9i)16-s + (−0.601 + 1.04i)17-s + (−6.46 + 3.73i)19-s + (3.21 + 5.56i)20-s + (3.36 − 5.83i)22-s + 3.04i·23-s − 3.55·25-s + (−2.55 − 4.42i)26-s + ⋯ |
L(s) = 1 | + (−1.65 − 0.957i)2-s + (1.33 + 2.31i)4-s + 0.537·5-s − 3.19i·8-s + (−0.892 − 0.515i)10-s + 0.749i·11-s + (0.452 + 0.261i)13-s + (−1.72 + 2.99i)16-s + (−0.145 + 0.252i)17-s + (−1.48 + 0.856i)19-s + (0.717 + 1.24i)20-s + (0.718 − 1.24i)22-s + 0.634i·23-s − 0.710·25-s + (−0.500 − 0.867i)26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0832 - 0.996i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0832 - 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.3453244029\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.3453244029\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 7 | \( 1 \) |
good | 2 | \( 1 + (2.34 + 1.35i)T + (1 + 1.73i)T^{2} \) |
| 5 | \( 1 - 1.20T + 5T^{2} \) |
| 11 | \( 1 - 2.48iT - 11T^{2} \) |
| 13 | \( 1 + (-1.63 - 0.942i)T + (6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 + (0.601 - 1.04i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (6.46 - 3.73i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 - 3.04iT - 23T^{2} \) |
| 29 | \( 1 + (-0.173 + 0.100i)T + (14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (-3.03 + 1.75i)T + (15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (0.865 + 1.49i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (3.36 - 5.82i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (-0.00656 - 0.0113i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (-0.717 + 1.24i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (8.58 + 4.95i)T + (26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (6.10 + 10.5i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (9.73 + 5.62i)T + (30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-2.57 - 4.46i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 - 12.0iT - 71T^{2} \) |
| 73 | \( 1 + (7.51 + 4.33i)T + (36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (2.74 - 4.75i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (1.60 + 2.78i)T + (-41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + (3.98 + 6.89i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (-2.06 + 1.19i)T + (48.5 - 84.0i)T^{2} \) |
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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
−9.848891909893762561620249051606, −9.249461981562054810829243644854, −8.362734051487394546050718214687, −7.83921660305713200805116159555, −6.79821240392743005842018867808, −6.07324597030196889693259736330, −4.38931835033245413572963162043, −3.39888268803495135798055173843, −2.13990190366062183843677542544, −1.56471900366141507989201108767,
0.24116064665156245399068416654, 1.59541359222922364491488385160, 2.74607970917816640694644349333, 4.61495999112352646683248007088, 5.80608699659539759550424781851, 6.25724883297710631227327568936, 7.03966298042798544748687207705, 7.982007076064721472656320784530, 8.702996842115368584159040646259, 9.103334695271131446845161633139