L(s) = 1 | + (−0.620 − 1.79i)2-s + (−0.888 − 0.458i)3-s + (−1.26 + 0.991i)4-s + (−3.16 − 0.302i)5-s + (−0.270 + 1.87i)6-s + (1.40 − 2.24i)7-s + (−0.632 − 0.406i)8-s + (0.580 + 0.814i)9-s + (1.42 + 5.87i)10-s + (−0.842 + 2.43i)11-s + (1.57 − 0.303i)12-s + (−3.50 − 1.02i)13-s + (−4.89 − 1.11i)14-s + (2.67 + 1.72i)15-s + (−1.09 + 4.50i)16-s + (−1.43 + 0.574i)17-s + ⋯ |
L(s) = 1 | + (−0.439 − 1.26i)2-s + (−0.513 − 0.264i)3-s + (−0.630 + 0.495i)4-s + (−1.41 − 0.135i)5-s + (−0.110 + 0.767i)6-s + (0.529 − 0.848i)7-s + (−0.223 − 0.143i)8-s + (0.193 + 0.271i)9-s + (0.450 + 1.85i)10-s + (−0.254 + 0.734i)11-s + (0.454 − 0.0876i)12-s + (−0.970 − 0.285i)13-s + (−1.30 − 0.298i)14-s + (0.691 + 0.444i)15-s + (−0.273 + 1.12i)16-s + (−0.347 + 0.139i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.676 - 0.736i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.676 - 0.736i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.0275605 + 0.0121109i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.0275605 + 0.0121109i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (0.888 + 0.458i)T \) |
| 7 | \( 1 + (-1.40 + 2.24i)T \) |
| 23 | \( 1 + (-2.95 + 3.77i)T \) |
good | 2 | \( 1 + (0.620 + 1.79i)T + (-1.57 + 1.23i)T^{2} \) |
| 5 | \( 1 + (3.16 + 0.302i)T + (4.90 + 0.946i)T^{2} \) |
| 11 | \( 1 + (0.842 - 2.43i)T + (-8.64 - 6.79i)T^{2} \) |
| 13 | \( 1 + (3.50 + 1.02i)T + (10.9 + 7.02i)T^{2} \) |
| 17 | \( 1 + (1.43 - 0.574i)T + (12.3 - 11.7i)T^{2} \) |
| 19 | \( 1 + (-2.71 - 1.08i)T + (13.7 + 13.1i)T^{2} \) |
| 29 | \( 1 + (1.35 - 9.39i)T + (-27.8 - 8.17i)T^{2} \) |
| 31 | \( 1 + (0.0236 - 0.497i)T + (-30.8 - 2.94i)T^{2} \) |
| 37 | \( 1 + (-4.33 - 6.08i)T + (-12.1 + 34.9i)T^{2} \) |
| 41 | \( 1 + (3.04 + 6.66i)T + (-26.8 + 30.9i)T^{2} \) |
| 43 | \( 1 + (-2.95 + 1.90i)T + (17.8 - 39.1i)T^{2} \) |
| 47 | \( 1 + (4.58 - 7.94i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (5.25 + 5.01i)T + (2.52 + 52.9i)T^{2} \) |
| 59 | \( 1 + (0.828 + 3.41i)T + (-52.4 + 27.0i)T^{2} \) |
| 61 | \( 1 + (8.94 - 4.61i)T + (35.3 - 49.6i)T^{2} \) |
| 67 | \( 1 + (4.79 + 0.924i)T + (62.2 + 24.9i)T^{2} \) |
| 71 | \( 1 + (0.887 - 1.02i)T + (-10.1 - 70.2i)T^{2} \) |
| 73 | \( 1 + (6.13 - 4.82i)T + (17.2 - 70.9i)T^{2} \) |
| 79 | \( 1 + (5.24 - 5.00i)T + (3.75 - 78.9i)T^{2} \) |
| 83 | \( 1 + (0.967 - 2.11i)T + (-54.3 - 62.7i)T^{2} \) |
| 89 | \( 1 + (-0.309 - 6.49i)T + (-88.5 + 8.45i)T^{2} \) |
| 97 | \( 1 + (6.82 + 14.9i)T + (-63.5 + 73.3i)T^{2} \) |
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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
−11.03439406681016985928219084266, −10.58037938250127887959311227364, −9.637206011822944718243774313298, −8.430489353816393874511309227392, −7.53467299595756930417174819154, −6.86023896590318317985050378367, −4.99156898265289884903170215194, −4.19309518096601932011764994837, −2.98325596877707234206080974649, −1.37005876036602198327076222108,
0.02384123715792150667715563545, 2.91709668508307805146275761243, 4.46455396262396143452235864821, 5.36739080701746643801579670279, 6.29052567316942943369244347089, 7.49266157491285824589752031803, 7.82824454540406963338096669486, 8.851683304494790841259903497203, 9.624069096864720491744947811065, 11.19953292579725291396068381539