L(s) = 1 | + (−1.82 + 1.05i)2-s − 2.26·3-s + (1.22 − 2.12i)4-s + (−3.11 − 1.80i)5-s + (4.13 − 2.38i)6-s + 0.948i·8-s + 2.11·9-s + 7.59·10-s − 0.886i·11-s + (−2.76 + 4.79i)12-s + (1.17 − 3.40i)13-s + (7.05 + 4.07i)15-s + (1.44 + 2.51i)16-s + (2.48 − 4.29i)17-s + (−3.86 + 2.23i)18-s − 2.37i·19-s + ⋯ |
L(s) = 1 | + (−1.29 + 0.745i)2-s − 1.30·3-s + (0.612 − 1.06i)4-s + (−1.39 − 0.805i)5-s + (1.68 − 0.973i)6-s + 0.335i·8-s + 0.705·9-s + 2.40·10-s − 0.267i·11-s + (−0.799 + 1.38i)12-s + (0.325 − 0.945i)13-s + (1.82 + 1.05i)15-s + (0.362 + 0.627i)16-s + (0.601 − 1.04i)17-s + (−0.910 + 0.525i)18-s − 0.545i·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.997 - 0.0651i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.997 - 0.0651i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.00246577 + 0.0756326i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.00246577 + 0.0756326i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 7 | \( 1 \) |
| 13 | \( 1 + (-1.17 + 3.40i)T \) |
good | 2 | \( 1 + (1.82 - 1.05i)T + (1 - 1.73i)T^{2} \) |
| 3 | \( 1 + 2.26T + 3T^{2} \) |
| 5 | \( 1 + (3.11 + 1.80i)T + (2.5 + 4.33i)T^{2} \) |
| 11 | \( 1 + 0.886iT - 11T^{2} \) |
| 17 | \( 1 + (-2.48 + 4.29i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + 2.37iT - 19T^{2} \) |
| 23 | \( 1 + (1.92 + 3.34i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (0.640 - 1.11i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (7.33 - 4.23i)T + (15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (-8.34 + 4.81i)T + (18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (10.4 + 6.04i)T + (20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (1.82 + 3.15i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (2.58 + 1.49i)T + (23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (2.46 + 4.26i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (-6.34 - 3.66i)T + (29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 - 1.53T + 61T^{2} \) |
| 67 | \( 1 + 8.42iT - 67T^{2} \) |
| 71 | \( 1 + (5.58 - 3.22i)T + (35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + (6.19 - 3.57i)T + (36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (0.378 - 0.656i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 - 4.76iT - 83T^{2} \) |
| 89 | \( 1 + (-3.13 + 1.80i)T + (44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (-0.401 + 0.231i)T + (48.5 - 84.0i)T^{2} \) |
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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
−10.15953112642225937518494405104, −8.994260137283482704101689011961, −8.363493855745299046790974996489, −7.53829874042115436121686931945, −6.84376642662151660644641017035, −5.67977847063941072045691647880, −4.94339718376539608044709208771, −3.60486671046944158580367125217, −0.841040484710832615602534992357, −0.11600703205403235142566166106,
1.55796398827128677551117613758, 3.28698326564594000462381065060, 4.33954736376678819303103048854, 5.79423336507909861301467446763, 6.78189858553128348003727091602, 7.72215616594144008015394793841, 8.331055150122133309890675107964, 9.604625354969676587774933778859, 10.36284467320555436517161857267, 11.06444085896569715027524844660