L(s) = 1 | + (−1.06 − 0.617i)2-s + (−1.23 − 2.14i)4-s + (4.08 − 7.08i)5-s − 10.4·7-s + 7.99i·8-s + (−8.73 + 5.04i)10-s + 3.90·11-s + (−2.02 + 1.16i)13-s + (11.1 + 6.46i)14-s + (−0.0227 + 0.0394i)16-s + (−2.13 + 3.70i)17-s + (−17.4 + 7.45i)19-s − 20.2·20-s + (−4.17 − 2.40i)22-s + (−20.0 − 34.7i)23-s + ⋯ |
L(s) = 1 | + (−0.534 − 0.308i)2-s + (−0.309 − 0.536i)4-s + (0.817 − 1.41i)5-s − 1.49·7-s + 0.999i·8-s + (−0.873 + 0.504i)10-s + 0.354·11-s + (−0.155 + 0.0898i)13-s + (0.799 + 0.461i)14-s + (−0.00142 + 0.00246i)16-s + (−0.125 + 0.217i)17-s + (−0.919 + 0.392i)19-s − 1.01·20-s + (−0.189 − 0.109i)22-s + (−0.872 − 1.51i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 171 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 - 0.0226i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 171 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.999 - 0.0226i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(0.00660751 + 0.584240i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.00660751 + 0.584240i\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 19 | \( 1 + (17.4 - 7.45i)T \) |
good | 2 | \( 1 + (1.06 + 0.617i)T + (2 + 3.46i)T^{2} \) |
| 5 | \( 1 + (-4.08 + 7.08i)T + (-12.5 - 21.6i)T^{2} \) |
| 7 | \( 1 + 10.4T + 49T^{2} \) |
| 11 | \( 1 - 3.90T + 121T^{2} \) |
| 13 | \( 1 + (2.02 - 1.16i)T + (84.5 - 146. i)T^{2} \) |
| 17 | \( 1 + (2.13 - 3.70i)T + (-144.5 - 250. i)T^{2} \) |
| 23 | \( 1 + (20.0 + 34.7i)T + (-264.5 + 458. i)T^{2} \) |
| 29 | \( 1 + (10.6 - 6.17i)T + (420.5 - 728. i)T^{2} \) |
| 31 | \( 1 - 22.2iT - 961T^{2} \) |
| 37 | \( 1 + 62.2iT - 1.36e3T^{2} \) |
| 41 | \( 1 + (-44.7 - 25.8i)T + (840.5 + 1.45e3i)T^{2} \) |
| 43 | \( 1 + (-6.67 + 11.5i)T + (-924.5 - 1.60e3i)T^{2} \) |
| 47 | \( 1 + (24.1 + 41.8i)T + (-1.10e3 + 1.91e3i)T^{2} \) |
| 53 | \( 1 + (-67.7 + 39.1i)T + (1.40e3 - 2.43e3i)T^{2} \) |
| 59 | \( 1 + (82.7 + 47.7i)T + (1.74e3 + 3.01e3i)T^{2} \) |
| 61 | \( 1 + (-18.8 - 32.7i)T + (-1.86e3 + 3.22e3i)T^{2} \) |
| 67 | \( 1 + (52.6 - 30.3i)T + (2.24e3 - 3.88e3i)T^{2} \) |
| 71 | \( 1 + (-78.8 - 45.5i)T + (2.52e3 + 4.36e3i)T^{2} \) |
| 73 | \( 1 + (-41.7 + 72.3i)T + (-2.66e3 - 4.61e3i)T^{2} \) |
| 79 | \( 1 + (-0.466 - 0.269i)T + (3.12e3 + 5.40e3i)T^{2} \) |
| 83 | \( 1 - 54.0T + 6.88e3T^{2} \) |
| 89 | \( 1 + (-81.6 + 47.1i)T + (3.96e3 - 6.85e3i)T^{2} \) |
| 97 | \( 1 + (126. + 73.2i)T + (4.70e3 + 8.14e3i)T^{2} \) |
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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
−12.33136155246207814018719961374, −10.63624310786190004125723550740, −9.827614984509967780167640870742, −9.144259367247533067808997578189, −8.449773041515736134688618132885, −6.44158181787951043739244953066, −5.60369901948174533915352557726, −4.26571464315779009862996505949, −2.05800577660206821932560172786, −0.40721440269600467823344194697,
2.72130761790985842943513473213, 3.79778835146608226904934236713, 6.09135242736894910363109220700, 6.72353193848163237240432637227, 7.71080741017069562013108333078, 9.350734304050331925641201678394, 9.700984970611877117871195465692, 10.74493810104503801831529742560, 12.09949393586050055489745615400, 13.26792979186973659899426805275