| L(s) = 1 | + (1.42 − 0.990i)3-s + (−2.13 + 1.79i)5-s + (−2.82 + 1.02i)7-s + (1.03 − 2.81i)9-s + (−0.488 − 0.409i)11-s + (−1.01 − 5.75i)13-s + (−1.25 + 4.66i)15-s + (−0.343 − 0.595i)17-s + (−0.768 + 1.33i)19-s + (−2.99 + 4.26i)21-s + (−8.70 − 3.16i)23-s + (0.480 − 2.72i)25-s + (−1.31 − 5.02i)27-s + (−0.568 + 3.22i)29-s + (0.333 + 0.121i)31-s + ⋯ |
| L(s) = 1 | + (0.820 − 0.571i)3-s + (−0.954 + 0.801i)5-s + (−1.06 + 0.389i)7-s + (0.345 − 0.938i)9-s + (−0.147 − 0.123i)11-s + (−0.281 − 1.59i)13-s + (−0.324 + 1.20i)15-s + (−0.0834 − 0.144i)17-s + (−0.176 + 0.305i)19-s + (−0.654 + 0.931i)21-s + (−1.81 − 0.660i)23-s + (0.0961 − 0.545i)25-s + (−0.253 − 0.967i)27-s + (−0.105 + 0.598i)29-s + (0.0599 + 0.0218i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 864 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.931 + 0.363i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 864 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.931 + 0.363i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.0857649 - 0.455311i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0857649 - 0.455311i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (-1.42 + 0.990i)T \) |
| good | 5 | \( 1 + (2.13 - 1.79i)T + (0.868 - 4.92i)T^{2} \) |
| 7 | \( 1 + (2.82 - 1.02i)T + (5.36 - 4.49i)T^{2} \) |
| 11 | \( 1 + (0.488 + 0.409i)T + (1.91 + 10.8i)T^{2} \) |
| 13 | \( 1 + (1.01 + 5.75i)T + (-12.2 + 4.44i)T^{2} \) |
| 17 | \( 1 + (0.343 + 0.595i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (0.768 - 1.33i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (8.70 + 3.16i)T + (17.6 + 14.7i)T^{2} \) |
| 29 | \( 1 + (0.568 - 3.22i)T + (-27.2 - 9.91i)T^{2} \) |
| 31 | \( 1 + (-0.333 - 0.121i)T + (23.7 + 19.9i)T^{2} \) |
| 37 | \( 1 + (3.90 + 6.76i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (-0.503 - 2.85i)T + (-38.5 + 14.0i)T^{2} \) |
| 43 | \( 1 + (8.93 + 7.49i)T + (7.46 + 42.3i)T^{2} \) |
| 47 | \( 1 + (2.06 - 0.752i)T + (36.0 - 30.2i)T^{2} \) |
| 53 | \( 1 - 9.65T + 53T^{2} \) |
| 59 | \( 1 + (7.72 - 6.48i)T + (10.2 - 58.1i)T^{2} \) |
| 61 | \( 1 + (-11.7 + 4.29i)T + (46.7 - 39.2i)T^{2} \) |
| 67 | \( 1 + (1.44 + 8.16i)T + (-62.9 + 22.9i)T^{2} \) |
| 71 | \( 1 + (-7.06 - 12.2i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (4.61 - 7.99i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (0.0301 - 0.170i)T + (-74.2 - 27.0i)T^{2} \) |
| 83 | \( 1 + (-0.386 + 2.19i)T + (-77.9 - 28.3i)T^{2} \) |
| 89 | \( 1 + (4.65 - 8.06i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (0.322 + 0.270i)T + (16.8 + 95.5i)T^{2} \) |
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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
−9.930073529741044020596397046173, −8.717956060429810737597564321596, −8.053703787647094381418851092624, −7.32178696520257593291159553820, −6.54833828305804884104368960827, −5.60157391677050279892040235331, −3.85465858060423588217341025347, −3.24978399329092660828101853822, −2.37492871916307793109181793496, −0.18833948073633474922810005893,
1.98673473174569055972516208804, 3.45904896562859133435055506727, 4.13038016377387011948635517117, 4.83353643555105986342646148010, 6.34672030156631924066832223337, 7.30088965684349771683700102751, 8.154328596729734932435956483119, 8.848333618238153060083712356169, 9.716886050462882981786335217447, 10.14843996399735975577723724829
