L(s) = 1 | + (0.5 − 0.866i)2-s + (−0.499 − 0.866i)4-s + (−2.11 + 0.712i)5-s + (1.63 + 2.07i)7-s − 0.999·8-s + (−0.442 + 2.19i)10-s + (5.48 − 3.16i)11-s + 1.05·13-s + (2.61 − 0.381i)14-s + (−0.5 + 0.866i)16-s + (−4.01 + 2.31i)17-s + (5.35 + 3.08i)19-s + (1.67 + 1.47i)20-s − 6.33i·22-s + (3.52 − 6.10i)23-s + ⋯ |
L(s) = 1 | + (0.353 − 0.612i)2-s + (−0.249 − 0.433i)4-s + (−0.947 + 0.318i)5-s + (0.619 + 0.784i)7-s − 0.353·8-s + (−0.139 + 0.693i)10-s + (1.65 − 0.954i)11-s + 0.292·13-s + (0.699 − 0.101i)14-s + (−0.125 + 0.216i)16-s + (−0.973 + 0.562i)17-s + (1.22 + 0.708i)19-s + (0.374 + 0.330i)20-s − 1.34i·22-s + (0.734 − 1.27i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.817 + 0.575i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.817 + 0.575i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.64006 - 0.518988i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.64006 - 0.518988i\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + (2.11 - 0.712i)T \) |
| 7 | \( 1 + (-1.63 - 2.07i)T \) |
good | 11 | \( 1 + (-5.48 + 3.16i)T + (5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 - 1.05T + 13T^{2} \) |
| 17 | \( 1 + (4.01 - 2.31i)T + (8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-5.35 - 3.08i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-3.52 + 6.10i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 - 2.98iT - 29T^{2} \) |
| 31 | \( 1 + (-5.46 + 3.15i)T + (15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (-3.01 - 1.73i)T + (18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 - 6.97T + 41T^{2} \) |
| 43 | \( 1 - 2.58iT - 43T^{2} \) |
| 47 | \( 1 + (7.06 + 4.07i)T + (23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (4.43 + 7.68i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (0.452 + 0.783i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (8.81 + 5.08i)T + (30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (8.51 - 4.91i)T + (33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 - 14.4iT - 71T^{2} \) |
| 73 | \( 1 + (-4.18 - 7.24i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (-2.73 + 4.73i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 - 14.6iT - 83T^{2} \) |
| 89 | \( 1 + (1.91 - 3.32i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + 5.87T + 97T^{2} \) |
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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
−11.02509900459210487573593977309, −9.648954389076695122383200364345, −8.690342609202577434714344112503, −8.199806455360213782839188598200, −6.74731983205725300627841666390, −5.99268032635823112509640120997, −4.68346104219622174357916333155, −3.83171792176990323065626075809, −2.83112550129946818351982324986, −1.21509511055922944371731920691,
1.19625594249307466173763385488, 3.34425468614608197022460613358, 4.41825835087594072017403376534, 4.79620452679921201716485997630, 6.38075490683359831765177707506, 7.30508795256080938425712587316, 7.66178381698246796159848400341, 8.986833769577174903839739106353, 9.437457329448181630137312571452, 10.98386058030686502899249455966