L(s) = 1 | + (−1.84 − 1.06i)2-s + (0.0894 + 0.154i)3-s + (1.25 + 2.18i)4-s + (3.12 + 1.80i)5-s − 0.380i·6-s + (−1.20 − 2.35i)7-s − 1.10i·8-s + (1.48 − 2.57i)9-s + (−3.83 − 6.63i)10-s + (3.45 − 1.99i)11-s + (−0.225 + 0.389i)12-s + (−2.51 + 2.58i)13-s + (−0.274 + 5.61i)14-s + 0.644i·15-s + (1.34 − 2.33i)16-s + (2.39 + 4.14i)17-s + ⋯ |
L(s) = 1 | + (−1.30 − 0.751i)2-s + (0.0516 + 0.0894i)3-s + (0.629 + 1.09i)4-s + (1.39 + 0.806i)5-s − 0.155i·6-s + (−0.457 − 0.889i)7-s − 0.389i·8-s + (0.494 − 0.856i)9-s + (−1.21 − 2.09i)10-s + (1.04 − 0.601i)11-s + (−0.0649 + 0.112i)12-s + (−0.698 + 0.715i)13-s + (−0.0734 + 1.50i)14-s + 0.166i·15-s + (0.337 − 0.583i)16-s + (0.580 + 1.00i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.725 + 0.687i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.725 + 0.687i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.612732 - 0.244207i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.612732 - 0.244207i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 7 | \( 1 + (1.20 + 2.35i)T \) |
| 13 | \( 1 + (2.51 - 2.58i)T \) |
good | 2 | \( 1 + (1.84 + 1.06i)T + (1 + 1.73i)T^{2} \) |
| 3 | \( 1 + (-0.0894 - 0.154i)T + (-1.5 + 2.59i)T^{2} \) |
| 5 | \( 1 + (-3.12 - 1.80i)T + (2.5 + 4.33i)T^{2} \) |
| 11 | \( 1 + (-3.45 + 1.99i)T + (5.5 - 9.52i)T^{2} \) |
| 17 | \( 1 + (-2.39 - 4.14i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (2.72 + 1.57i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (1.08 - 1.88i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + 6.57T + 29T^{2} \) |
| 31 | \( 1 + (1.28 - 0.743i)T + (15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (4.29 + 2.48i)T + (18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 - 2.11iT - 41T^{2} \) |
| 43 | \( 1 + 1.43T + 43T^{2} \) |
| 47 | \( 1 + (0.882 + 0.509i)T + (23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (3.01 + 5.22i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (4.24 - 2.45i)T + (29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-1.01 + 1.76i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (3.38 - 1.95i)T + (33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 - 8.80iT - 71T^{2} \) |
| 73 | \( 1 + (-2.67 + 1.54i)T + (36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (0.984 - 1.70i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + 7.66iT - 83T^{2} \) |
| 89 | \( 1 + (-11.0 - 6.39i)T + (44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + 1.35iT - 97T^{2} \) |
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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
−14.04212975521038480069498777590, −12.74351702762822507406367548345, −11.38861075850204996375986151582, −10.35784059588994108982980718737, −9.726468790446003095808358800835, −9.006462443755232010282653093071, −7.17694468674694202104770265600, −6.20541103340616928723532461626, −3.54575165461559176440033136269, −1.69598772035101159798568284621,
1.88202770229691155814512467317, 5.18172163762859265146384383811, 6.31291709295714839012282770288, 7.59454722222392282441876403191, 8.908762612417814331517389350583, 9.565580272298623974391979216417, 10.26575057773635245458946678884, 12.29866846342312096083651902132, 13.09975648704417748513164148622, 14.43220721470661057712172168383