L(s) = 1 | + (−0.813 − 0.813i)2-s + (2.01 − 2.01i)3-s − 0.675i·4-s − 3.28·6-s + (2.67 − 2.67i)7-s + (−2.17 + 2.17i)8-s − 5.15i·9-s + 2.15·11-s + (−1.36 − 1.36i)12-s + (−2.01 + 2.01i)13-s − 4.35·14-s + 2.19·16-s + (−1.80 + 1.80i)17-s + (−4.19 + 4.19i)18-s + (4.35 − 0.193i)19-s + ⋯ |
L(s) = 1 | + (−0.575 − 0.575i)2-s + (1.16 − 1.16i)3-s − 0.337i·4-s − 1.34·6-s + (1.01 − 1.01i)7-s + (−0.769 + 0.769i)8-s − 1.71i·9-s + 0.650·11-s + (−0.393 − 0.393i)12-s + (−0.560 + 0.560i)13-s − 1.16·14-s + 0.548·16-s + (−0.438 + 0.438i)17-s + (−0.989 + 0.989i)18-s + (0.999 − 0.0444i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.826 + 0.563i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.826 + 0.563i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.481189 - 1.56085i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.481189 - 1.56085i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 \) |
| 19 | \( 1 + (-4.35 + 0.193i)T \) |
good | 2 | \( 1 + (0.813 + 0.813i)T + 2iT^{2} \) |
| 3 | \( 1 + (-2.01 + 2.01i)T - 3iT^{2} \) |
| 7 | \( 1 + (-2.67 + 2.67i)T - 7iT^{2} \) |
| 11 | \( 1 - 2.15T + 11T^{2} \) |
| 13 | \( 1 + (2.01 - 2.01i)T - 13iT^{2} \) |
| 17 | \( 1 + (1.80 - 1.80i)T - 17iT^{2} \) |
| 23 | \( 1 + (-2.67 - 2.67i)T + 23iT^{2} \) |
| 29 | \( 1 + 7.29T + 29T^{2} \) |
| 31 | \( 1 - 2.09iT - 31T^{2} \) |
| 37 | \( 1 + (-0.391 - 0.391i)T + 37iT^{2} \) |
| 41 | \( 1 - 3.51iT - 41T^{2} \) |
| 43 | \( 1 + (-5.24 - 5.24i)T + 43iT^{2} \) |
| 47 | \( 1 + (1.63 - 1.63i)T - 47iT^{2} \) |
| 53 | \( 1 + (4.43 - 4.43i)T - 53iT^{2} \) |
| 59 | \( 1 + 3.51T + 59T^{2} \) |
| 61 | \( 1 + 4.93T + 61T^{2} \) |
| 67 | \( 1 + (7.68 + 7.68i)T + 67iT^{2} \) |
| 71 | \( 1 + 10.8iT - 71T^{2} \) |
| 73 | \( 1 + (9.73 + 9.73i)T + 73iT^{2} \) |
| 79 | \( 1 - 5.19T + 79T^{2} \) |
| 83 | \( 1 + (-8.44 - 8.44i)T + 83iT^{2} \) |
| 89 | \( 1 - 5.60T + 89T^{2} \) |
| 97 | \( 1 + (-5.59 - 5.59i)T + 97iT^{2} \) |
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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
−10.72557965238317958254209478882, −9.422103457152305801930562463699, −9.048028120037029062052143175303, −7.84909100617430221580179380836, −7.39730066055765411756777799854, −6.27663108769124582801573011898, −4.71199655715028510400329280736, −3.26652989929715810538784665368, −1.88420899210643139109019360402, −1.22664173459353405325987202266,
2.40828679500271495065038941645, 3.44051367281496132874004090384, 4.58088755344374605998687000416, 5.62567215439025953977828293204, 7.25233650992303655073187587791, 7.977974338496672757190988157560, 8.906682618928844393097274426903, 9.122958555510987797688112286982, 10.05538654438885118860143138402, 11.27723745713784457912881964656