L(s) = 1 | + (−0.5 − 0.866i)2-s + (−0.514 − 1.65i)3-s + (−0.499 + 0.866i)4-s − 3.58i·5-s + (−1.17 + 1.27i)6-s + (−1.90 − 1.83i)7-s + 0.999·8-s + (−2.47 + 1.70i)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 + (−5.92 + 1.84i)15-s + (−0.5 − 0.866i)16-s + (−1.57 + 2.73i)17-s + ⋯ |
L(s) = 1 | + (−0.353 − 0.612i)2-s + (−0.296 − 0.954i)3-s + (−0.249 + 0.433i)4-s − 1.60i·5-s + (−0.479 + 0.519i)6-s + (−0.718 − 0.695i)7-s + 0.353·8-s + (−0.823 + 0.566i)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 + (−1.53 + 0.475i)15-s + (−0.125 − 0.216i)16-s + (−0.383 + 0.663i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.635 - 0.772i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.635 - 0.772i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.306493 + 0.648816i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.306493 + 0.648816i\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + (0.514 + 1.65i)T \) |
| 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
−10.44767639797359603364924304096, −9.127278742118728908881722317135, −8.569106106482696152260735465383, −7.78695042631088741625371320076, −6.62349950104844951054540796503, −5.58304894967179253893550018921, −4.48101778415906144095167380546, −3.15290192787074348271991569341, −1.45725283286224162503717187878, −0.51139940106187395401655902400,
2.66554164646466433101195234120, 3.62984454973125416058124461305, 4.99704857063538056482495294491, 6.16375720655870739375230999762, 6.60328115456517644025815686269, 7.66193795582795167546147936703, 9.079832958345395636966564435921, 9.477387596070722518851246508481, 10.42933496876006263163834765552, 11.14790094737611860328435273108