L(s) = 1 | + (−0.5 + 0.866i)2-s + (−0.499 − 0.866i)4-s + (−2.11 − 0.728i)5-s + (2.30 − 1.29i)7-s + 0.999·8-s + (1.68 − 1.46i)10-s + (−1.11 + 0.641i)11-s − 6.14·13-s + (−0.0298 + 2.64i)14-s + (−0.5 + 0.866i)16-s + (−5.64 + 3.26i)17-s + (5.22 + 3.01i)19-s + (0.426 + 2.19i)20-s − 1.28i·22-s + (−1.43 + 2.49i)23-s + ⋯ |
L(s) = 1 | + (−0.353 + 0.612i)2-s + (−0.249 − 0.433i)4-s + (−0.945 − 0.325i)5-s + (0.871 − 0.490i)7-s + 0.353·8-s + (0.533 − 0.463i)10-s + (−0.335 + 0.193i)11-s − 1.70·13-s + (−0.00798 + 0.707i)14-s + (−0.125 + 0.216i)16-s + (−1.37 + 0.790i)17-s + (1.19 + 0.692i)19-s + (0.0953 + 0.490i)20-s − 0.273i·22-s + (−0.300 + 0.519i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.989 + 0.145i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.989 + 0.145i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.0107966 - 0.147823i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.0107966 - 0.147823i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.5 - 0.866i)T \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (2.11 + 0.728i)T \) |
| 7 | \( 1 + (-2.30 + 1.29i)T \) |
good | 11 | \( 1 + (1.11 - 0.641i)T + (5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + 6.14T + 13T^{2} \) |
| 17 | \( 1 + (5.64 - 3.26i)T + (8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-5.22 - 3.01i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (1.43 - 2.49i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + 1.35iT - 29T^{2} \) |
| 31 | \( 1 + (7.49 - 4.32i)T + (15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (8.25 + 4.76i)T + (18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + 8.71T + 41T^{2} \) |
| 43 | \( 1 + 5.35iT - 43T^{2} \) |
| 47 | \( 1 + (-0.698 - 0.403i)T + (23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (-3.33 - 5.77i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (-0.798 - 1.38i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (5.50 + 3.17i)T + (30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (4.67 - 2.69i)T + (33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + 15.6iT - 71T^{2} \) |
| 73 | \( 1 + (-6.20 - 10.7i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (5.59 - 9.68i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + 3.74iT - 83T^{2} \) |
| 89 | \( 1 + (1.81 - 3.15i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 - 8.76T + 97T^{2} \) |
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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
−10.91472803174505847830758233961, −10.18947589977345146509737735398, −9.104355158572010231763032210159, −8.294215508703426495128393178574, −7.41667173686515700611344686714, −7.11507418547415407229349926450, −5.41655469794303777669889131944, −4.76605758041849071943036791929, −3.73013148414163065548707968195, −1.81980726278983370037476348589,
0.087040441079224513116139766274, 2.18141177492651103074309472833, 3.11807446716457654465666545318, 4.57566630356025004012131022907, 5.14315866038981227016440315409, 6.98176254042462511620826684536, 7.53658159968241317524131135688, 8.479126864567236072638415411467, 9.251830693646353498875810189961, 10.24772171739968851950161549391