L(s) = 1 | + (−0.414 − 0.414i)3-s − 2.65i·9-s + (−4.41 + 4.41i)11-s − 5.65·17-s + (5.24 + 5.24i)19-s + 5i·25-s + (−2.34 + 2.34i)27-s + 3.65·33-s − 6i·41-s + (−9.24 + 9.24i)43-s + 7·49-s + (2.34 + 2.34i)51-s − 4.34i·57-s + (−10.0 + 10.0i)59-s + (2.75 + 2.75i)67-s + ⋯ |
L(s) = 1 | + (−0.239 − 0.239i)3-s − 0.885i·9-s + (−1.33 + 1.33i)11-s − 1.37·17-s + (1.20 + 1.20i)19-s + i·25-s + (−0.450 + 0.450i)27-s + 0.636·33-s − 0.937i·41-s + (−1.40 + 1.40i)43-s + 49-s + (0.328 + 0.328i)51-s − 0.575i·57-s + (−1.31 + 1.31i)59-s + (0.336 + 0.336i)67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.382 - 0.923i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.382 - 0.923i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.6502851402\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6502851402\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
good | 3 | \( 1 + (0.414 + 0.414i)T + 3iT^{2} \) |
| 5 | \( 1 - 5iT^{2} \) |
| 7 | \( 1 - 7T^{2} \) |
| 11 | \( 1 + (4.41 - 4.41i)T - 11iT^{2} \) |
| 13 | \( 1 + 13iT^{2} \) |
| 17 | \( 1 + 5.65T + 17T^{2} \) |
| 19 | \( 1 + (-5.24 - 5.24i)T + 19iT^{2} \) |
| 23 | \( 1 - 23T^{2} \) |
| 29 | \( 1 + 29iT^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 - 37iT^{2} \) |
| 41 | \( 1 + 6iT - 41T^{2} \) |
| 43 | \( 1 + (9.24 - 9.24i)T - 43iT^{2} \) |
| 47 | \( 1 + 47T^{2} \) |
| 53 | \( 1 - 53iT^{2} \) |
| 59 | \( 1 + (10.0 - 10.0i)T - 59iT^{2} \) |
| 61 | \( 1 + 61iT^{2} \) |
| 67 | \( 1 + (-2.75 - 2.75i)T + 67iT^{2} \) |
| 71 | \( 1 - 71T^{2} \) |
| 73 | \( 1 - 16.9iT - 73T^{2} \) |
| 79 | \( 1 + 79T^{2} \) |
| 83 | \( 1 + (7.58 + 7.58i)T + 83iT^{2} \) |
| 89 | \( 1 - 5.65iT - 89T^{2} \) |
| 97 | \( 1 + 16.9T + 97T^{2} \) |
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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
−10.06863023930523732802803062902, −9.539772300198722957139308700564, −8.543575747329207558032421293984, −7.49280995378075245156668909699, −7.01126384634165123003215594472, −5.89916428668121256222192473734, −5.08484612656141040723452359098, −4.04605528702453039880497134791, −2.84178696095739283497759850503, −1.57642784869622628734582406800,
0.29313741563596812817287784154, 2.28998756619477056284438854666, 3.19946971412468669710405256459, 4.65362040263685293880417108502, 5.22345022923748934383368639555, 6.17481309967527955008673703619, 7.22283249456590831896030918300, 8.128631745379989583821837749745, 8.733379719437086568597191625201, 9.788739478242926212696272276202