L(s) = 1 | + (0.903 − 1.47i)3-s + (−1.26 − 3.46i)5-s + (1.82 − 0.321i)7-s + (−1.36 − 2.67i)9-s + (−2.49 − 0.907i)11-s + (4.15 + 3.48i)13-s + (−6.25 − 1.26i)15-s + (−5.06 − 2.92i)17-s + (6.05 − 3.49i)19-s + (1.17 − 2.98i)21-s + (0.349 − 1.98i)23-s + (−6.56 + 5.51i)25-s + (−5.18 − 0.397i)27-s + (4.64 + 5.53i)29-s + (−5.28 − 0.932i)31-s + ⋯ |
L(s) = 1 | + (0.521 − 0.852i)3-s + (−0.563 − 1.54i)5-s + (0.689 − 0.121i)7-s + (−0.455 − 0.890i)9-s + (−0.751 − 0.273i)11-s + (1.15 + 0.966i)13-s + (−1.61 − 0.327i)15-s + (−1.22 − 0.709i)17-s + (1.38 − 0.802i)19-s + (0.256 − 0.651i)21-s + (0.0728 − 0.413i)23-s + (−1.31 + 1.10i)25-s + (−0.997 − 0.0764i)27-s + (0.861 + 1.02i)29-s + (−0.950 − 0.167i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 864 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.829 + 0.558i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 864 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.829 + 0.558i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.461754 - 1.51264i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.461754 - 1.51264i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (-0.903 + 1.47i)T \) |
good | 5 | \( 1 + (1.26 + 3.46i)T + (-3.83 + 3.21i)T^{2} \) |
| 7 | \( 1 + (-1.82 + 0.321i)T + (6.57 - 2.39i)T^{2} \) |
| 11 | \( 1 + (2.49 + 0.907i)T + (8.42 + 7.07i)T^{2} \) |
| 13 | \( 1 + (-4.15 - 3.48i)T + (2.25 + 12.8i)T^{2} \) |
| 17 | \( 1 + (5.06 + 2.92i)T + (8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-6.05 + 3.49i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-0.349 + 1.98i)T + (-21.6 - 7.86i)T^{2} \) |
| 29 | \( 1 + (-4.64 - 5.53i)T + (-5.03 + 28.5i)T^{2} \) |
| 31 | \( 1 + (5.28 + 0.932i)T + (29.1 + 10.6i)T^{2} \) |
| 37 | \( 1 + (1.56 - 2.71i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (5.45 - 6.49i)T + (-7.11 - 40.3i)T^{2} \) |
| 43 | \( 1 + (-3.31 + 9.11i)T + (-32.9 - 27.6i)T^{2} \) |
| 47 | \( 1 + (-1.04 - 5.94i)T + (-44.1 + 16.0i)T^{2} \) |
| 53 | \( 1 - 2.61iT - 53T^{2} \) |
| 59 | \( 1 + (1.31 - 0.478i)T + (45.1 - 37.9i)T^{2} \) |
| 61 | \( 1 + (1.08 + 6.15i)T + (-57.3 + 20.8i)T^{2} \) |
| 67 | \( 1 + (-6.80 + 8.10i)T + (-11.6 - 65.9i)T^{2} \) |
| 71 | \( 1 + (-3.21 + 5.56i)T + (-35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + (0.960 + 1.66i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (-4.70 - 5.61i)T + (-13.7 + 77.7i)T^{2} \) |
| 83 | \( 1 + (-1.31 + 1.10i)T + (14.4 - 81.7i)T^{2} \) |
| 89 | \( 1 + (-8.51 + 4.91i)T + (44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (-5.07 - 1.84i)T + (74.3 + 62.3i)T^{2} \) |
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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
−9.282830168826925837561538434779, −8.875770979982648088859804116161, −8.237234424299296677233888075533, −7.45784491352339213855232046475, −6.52786028203789446202008716815, −5.19418534246229084250561056092, −4.55394571299977654921654615125, −3.29128045576855104486281733992, −1.79460725213394906197967069150, −0.73706037181548060804693294944,
2.20824436322942199080280517624, 3.27804076962516705376240463553, 3.89495626221040229060475195623, 5.16870713136889283673844601397, 6.10317991378312890550667067880, 7.36613812385216304602781386681, 7.983353265218958896010008239107, 8.665998872762638656477891067747, 9.958390394175607403707121815984, 10.52195418587668039044666736429