L(s) = 1 | + (0.5 + 0.866i)2-s + (1.17 + 1.27i)3-s + (−0.499 + 0.866i)4-s + 3.58i·5-s + (−0.514 + 1.65i)6-s + (−1.90 − 1.83i)7-s − 0.999·8-s + (−0.237 + 2.99i)9-s + (−3.10 + 1.79i)10-s + (0.630 + 1.09i)11-s + (−1.68 + 0.381i)12-s + (2.88 − 2.16i)13-s + (0.641 − 2.56i)14-s + (−4.55 + 4.21i)15-s + (−0.5 − 0.866i)16-s + (1.57 − 2.73i)17-s + ⋯ |
L(s) = 1 | + (0.353 + 0.612i)2-s + (0.678 + 0.734i)3-s + (−0.249 + 0.433i)4-s + 1.60i·5-s + (−0.209 + 0.675i)6-s + (−0.718 − 0.695i)7-s − 0.353·8-s + (−0.0791 + 0.996i)9-s + (−0.981 + 0.566i)10-s + (0.190 + 0.329i)11-s + (−0.487 + 0.110i)12-s + (0.799 − 0.600i)13-s + (0.171 − 0.686i)14-s + (−1.17 + 1.08i)15-s + (−0.125 − 0.216i)16-s + (0.383 − 0.663i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.905 - 0.424i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.905 - 0.424i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.411317 + 1.84710i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.411317 + 1.84710i\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + (-1.17 - 1.27i)T \) |
| 7 | \( 1 + (1.90 + 1.83i)T \) |
| 13 | \( 1 + (-2.88 + 2.16i)T \) |
good | 5 | \( 1 - 3.58iT - 5T^{2} \) |
| 11 | \( 1 + (-0.630 - 1.09i)T + (-5.5 + 9.52i)T^{2} \) |
| 17 | \( 1 + (-1.57 + 2.73i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (0.386 - 0.670i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (7.28 - 4.20i)T + (11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (-6.76 + 3.90i)T + (14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 - 7.70T + 31T^{2} \) |
| 37 | \( 1 + (0.424 - 0.244i)T + (18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (-1.17 + 0.676i)T + (20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (-0.125 + 0.216i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 - 10.9iT - 47T^{2} \) |
| 53 | \( 1 + 12.1iT - 53T^{2} \) |
| 59 | \( 1 + (-10.4 - 6.02i)T + (29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (3.27 + 1.88i)T + (30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (10.4 - 6.06i)T + (33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (-0.0759 + 0.131i)T + (-35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 - 10.3T + 73T^{2} \) |
| 79 | \( 1 - 5.53T + 79T^{2} \) |
| 83 | \( 1 - 3.23iT - 83T^{2} \) |
| 89 | \( 1 + (5.69 - 3.28i)T + (44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (-2.21 + 3.83i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−10.95187568844252805145789096115, −10.03991915382923886806591297517, −9.770198354380269012535167849389, −8.258298243443519281880711120234, −7.57135500897003687487214005267, −6.65338300092505791113797684191, −5.85013307767295265242774361894, −4.30490235251320340685303982109, −3.48088547514615877661698934323, −2.72184954039353522445924238745,
0.958372841616203436602286598709, 2.16305363957774407794126393732, 3.52046879683732716016453905722, 4.51763255072390567560652773928, 5.86076028640069643025044594648, 6.49665301327575245803990230735, 8.273298376828385608366143561769, 8.599607611976243490903430385320, 9.352460208988168275920412339300, 10.29315544531995410351375317934