L(s) = 1 | + (1 − 1.73i)2-s + (−1.99 − 3.46i)4-s + (−1 + 1.73i)5-s + (−6 + 3.46i)7-s − 7.99·8-s + (1.99 + 3.46i)10-s + (−6 + 3.46i)11-s + (−1 + 1.73i)13-s + 13.8i·14-s + (−8 + 13.8i)16-s − 10·17-s + 20.7i·19-s + 7.99·20-s + 13.8i·22-s + (−24 − 13.8i)23-s + ⋯ |
L(s) = 1 | + (0.5 − 0.866i)2-s + (−0.499 − 0.866i)4-s + (−0.200 + 0.346i)5-s + (−0.857 + 0.494i)7-s − 0.999·8-s + (0.199 + 0.346i)10-s + (−0.545 + 0.314i)11-s + (−0.0769 + 0.133i)13-s + 0.989i·14-s + (−0.5 + 0.866i)16-s − 0.588·17-s + 1.09i·19-s + 0.399·20-s + 0.629i·22-s + (−1.04 − 0.602i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.173 - 0.984i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.173 - 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(0.221450 + 0.263914i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.221450 + 0.263914i\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-1 + 1.73i)T \) |
| 3 | \( 1 \) |
good | 5 | \( 1 + (1 - 1.73i)T + (-12.5 - 21.6i)T^{2} \) |
| 7 | \( 1 + (6 - 3.46i)T + (24.5 - 42.4i)T^{2} \) |
| 11 | \( 1 + (6 - 3.46i)T + (60.5 - 104. i)T^{2} \) |
| 13 | \( 1 + (1 - 1.73i)T + (-84.5 - 146. i)T^{2} \) |
| 17 | \( 1 + 10T + 289T^{2} \) |
| 19 | \( 1 - 20.7iT - 361T^{2} \) |
| 23 | \( 1 + (24 + 13.8i)T + (264.5 + 458. i)T^{2} \) |
| 29 | \( 1 + (13 + 22.5i)T + (-420.5 + 728. i)T^{2} \) |
| 31 | \( 1 + (6 + 3.46i)T + (480.5 + 832. i)T^{2} \) |
| 37 | \( 1 - 26T + 1.36e3T^{2} \) |
| 41 | \( 1 + (-29 + 50.2i)T + (-840.5 - 1.45e3i)T^{2} \) |
| 43 | \( 1 + (42 - 24.2i)T + (924.5 - 1.60e3i)T^{2} \) |
| 47 | \( 1 + (60 - 34.6i)T + (1.10e3 - 1.91e3i)T^{2} \) |
| 53 | \( 1 - 74T + 2.80e3T^{2} \) |
| 59 | \( 1 + (78 + 45.0i)T + (1.74e3 + 3.01e3i)T^{2} \) |
| 61 | \( 1 + (13 + 22.5i)T + (-1.86e3 + 3.22e3i)T^{2} \) |
| 67 | \( 1 + (6 + 3.46i)T + (2.24e3 + 3.88e3i)T^{2} \) |
| 71 | \( 1 - 5.04e3T^{2} \) |
| 73 | \( 1 + 46T + 5.32e3T^{2} \) |
| 79 | \( 1 + (-102 + 58.8i)T + (3.12e3 - 5.40e3i)T^{2} \) |
| 83 | \( 1 + (42 - 24.2i)T + (3.44e3 - 5.96e3i)T^{2} \) |
| 89 | \( 1 + 82T + 7.92e3T^{2} \) |
| 97 | \( 1 + (1 + 1.73i)T + (-4.70e3 + 8.14e3i)T^{2} \) |
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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
−11.72961434489786157942847585850, −10.78402659816629405112350240089, −9.940984964279912904362777562420, −9.205668706343231434137846308469, −7.940534805112929077970162727799, −6.51150779221721557336402637020, −5.67412141302988541619372049614, −4.34647083206120352549128820763, −3.22786289856363845548582311928, −2.09493291240081836587441258874,
0.12925517965759771049272670754, 2.92711068849275110330179030511, 4.10245543068972086000529079829, 5.15759336281776685108476172514, 6.30318045579560529415228275301, 7.14482798578321806455600881390, 8.145003103636377732392346662195, 9.053910286263710746606062638145, 10.06030159042584875631355785039, 11.29387342887454806574607139596