L(s) = 1 | + (−0.603 − 2.64i)2-s + (−4.83 + 2.32i)4-s + (0.940 + 1.17i)5-s + (2.57 + 0.599i)7-s + (5.68 + 7.13i)8-s + (2.55 − 3.19i)10-s + (−0.818 − 3.58i)11-s + (−0.912 − 3.99i)13-s + (0.0297 − 7.17i)14-s + (8.74 − 10.9i)16-s + (−3.10 − 1.49i)17-s + 5.74·19-s + (−7.28 − 3.50i)20-s + (−8.99 + 4.33i)22-s + (3.61 − 1.74i)23-s + ⋯ |
L(s) = 1 | + (−0.426 − 1.87i)2-s + (−2.41 + 1.16i)4-s + (0.420 + 0.527i)5-s + (0.973 + 0.226i)7-s + (2.01 + 2.52i)8-s + (0.806 − 1.01i)10-s + (−0.246 − 1.08i)11-s + (−0.253 − 1.10i)13-s + (0.00794 − 1.91i)14-s + (2.18 − 2.74i)16-s + (−0.753 − 0.362i)17-s + 1.31·19-s + (−1.62 − 0.784i)20-s + (−1.91 + 0.923i)22-s + (0.753 − 0.362i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.806 + 0.591i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.806 + 0.591i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.341751 - 1.04373i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.341751 - 1.04373i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 7 | \( 1 + (-2.57 - 0.599i)T \) |
good | 2 | \( 1 + (0.603 + 2.64i)T + (-1.80 + 0.867i)T^{2} \) |
| 5 | \( 1 + (-0.940 - 1.17i)T + (-1.11 + 4.87i)T^{2} \) |
| 11 | \( 1 + (0.818 + 3.58i)T + (-9.91 + 4.77i)T^{2} \) |
| 13 | \( 1 + (0.912 + 3.99i)T + (-11.7 + 5.64i)T^{2} \) |
| 17 | \( 1 + (3.10 + 1.49i)T + (10.5 + 13.2i)T^{2} \) |
| 19 | \( 1 - 5.74T + 19T^{2} \) |
| 23 | \( 1 + (-3.61 + 1.74i)T + (14.3 - 17.9i)T^{2} \) |
| 29 | \( 1 + (-5.37 - 2.58i)T + (18.0 + 22.6i)T^{2} \) |
| 31 | \( 1 + 3.17T + 31T^{2} \) |
| 37 | \( 1 + (-0.851 - 0.410i)T + (23.0 + 28.9i)T^{2} \) |
| 41 | \( 1 + (-0.694 - 0.870i)T + (-9.12 + 39.9i)T^{2} \) |
| 43 | \( 1 + (-4.96 + 6.22i)T + (-9.56 - 41.9i)T^{2} \) |
| 47 | \( 1 + (-0.237 - 1.04i)T + (-42.3 + 20.3i)T^{2} \) |
| 53 | \( 1 + (-2.81 + 1.35i)T + (33.0 - 41.4i)T^{2} \) |
| 59 | \( 1 + (4.89 - 6.13i)T + (-13.1 - 57.5i)T^{2} \) |
| 61 | \( 1 + (3.90 + 1.87i)T + (38.0 + 47.6i)T^{2} \) |
| 67 | \( 1 + 6.17T + 67T^{2} \) |
| 71 | \( 1 + (3.76 - 1.81i)T + (44.2 - 55.5i)T^{2} \) |
| 73 | \( 1 + (-0.253 + 1.10i)T + (-65.7 - 31.6i)T^{2} \) |
| 79 | \( 1 + 15.6T + 79T^{2} \) |
| 83 | \( 1 + (2.63 - 11.5i)T + (-74.7 - 36.0i)T^{2} \) |
| 89 | \( 1 + (-2.12 + 9.29i)T + (-80.1 - 38.6i)T^{2} \) |
| 97 | \( 1 + 3.94T + 97T^{2} \) |
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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
−10.73651960368146472192405985903, −10.28426834560889459200742150393, −9.108480467640126588962954544036, −8.471243562698634222255881294311, −7.49213734175566169323073976807, −5.59149196886534013974434876828, −4.66311925512416057277794507633, −3.17863356427787112544861898586, −2.50597422389011072535658043125, −0.951271690156370169624900465704,
1.48488582153952194105881722953, 4.45055737339450941257251216319, 4.89717786710981796242077292541, 5.91982474264901005105234170008, 7.13494417835095684777396980915, 7.53911961945376647568879780124, 8.696570997458574074297408417111, 9.295474916083458728278849558019, 10.07315587092547095344894632388, 11.36838409054381916957882552128