L(s) = 1 | + (1 + 1.73i)2-s + (−1.99 + 3.46i)4-s + (−4 + 6.92i)5-s − 7.99·8-s − 15.9·10-s + (5 − 8.66i)13-s + (−8 − 13.8i)16-s − 16·17-s + (−15.9 − 27.7i)20-s + (−19.4 − 33.7i)25-s + 20·26-s + (20 + 34.6i)29-s + (15.9 − 27.7i)32-s + (−16 − 27.7i)34-s − 70·37-s + ⋯ |
L(s) = 1 | + (0.5 + 0.866i)2-s + (−0.499 + 0.866i)4-s + (−0.800 + 1.38i)5-s − 0.999·8-s − 1.59·10-s + (0.384 − 0.666i)13-s + (−0.5 − 0.866i)16-s − 0.941·17-s + (−0.799 − 1.38i)20-s + (−0.779 − 1.35i)25-s + 0.769·26-s + (0.689 + 1.19i)29-s + (0.499 − 0.866i)32-s + (−0.470 − 0.815i)34-s − 1.89·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.766 + 0.642i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.766 + 0.642i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(0.327742 - 0.900466i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.327742 - 0.900466i\) |
\(L(2)\) |
|
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 | \( 1 + (4 - 6.92i)T + (-12.5 - 21.6i)T^{2} \) |
| 7 | \( 1 + (24.5 - 42.4i)T^{2} \) |
| 11 | \( 1 + (60.5 - 104. i)T^{2} \) |
| 13 | \( 1 + (-5 + 8.66i)T + (-84.5 - 146. i)T^{2} \) |
| 17 | \( 1 + 16T + 289T^{2} \) |
| 19 | \( 1 - 361T^{2} \) |
| 23 | \( 1 + (264.5 + 458. i)T^{2} \) |
| 29 | \( 1 + (-20 - 34.6i)T + (-420.5 + 728. i)T^{2} \) |
| 31 | \( 1 + (480.5 + 832. i)T^{2} \) |
| 37 | \( 1 + 70T + 1.36e3T^{2} \) |
| 41 | \( 1 + (40 - 69.2i)T + (-840.5 - 1.45e3i)T^{2} \) |
| 43 | \( 1 + (924.5 - 1.60e3i)T^{2} \) |
| 47 | \( 1 + (1.10e3 - 1.91e3i)T^{2} \) |
| 53 | \( 1 - 56T + 2.80e3T^{2} \) |
| 59 | \( 1 + (1.74e3 + 3.01e3i)T^{2} \) |
| 61 | \( 1 + (-11 - 19.0i)T + (-1.86e3 + 3.22e3i)T^{2} \) |
| 67 | \( 1 + (2.24e3 + 3.88e3i)T^{2} \) |
| 71 | \( 1 - 5.04e3T^{2} \) |
| 73 | \( 1 - 110T + 5.32e3T^{2} \) |
| 79 | \( 1 + (3.12e3 - 5.40e3i)T^{2} \) |
| 83 | \( 1 + (3.44e3 - 5.96e3i)T^{2} \) |
| 89 | \( 1 + 160T + 7.92e3T^{2} \) |
| 97 | \( 1 + (-65 - 112. i)T + (-4.70e3 + 8.14e3i)T^{2} \) |
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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
−11.98636428030059413797197522445, −11.12027530804224371655854997908, −10.26350244793926142092367740105, −8.794368793698016836283302189183, −7.947767147872384359725612164932, −6.97973803770808077768758938712, −6.41472933347097982395915889757, −5.03819507206687820226401091361, −3.75164624348903047542013367409, −2.90429620806351092847395531762,
0.37919183467176667099697283802, 1.89116904659612649578775882992, 3.71672629168417051970807388874, 4.50801195833676593463465174585, 5.41921783526817406563375777299, 6.81358686711262502366751244350, 8.444717379838500010018585427938, 8.870837503566441489089058178462, 9.985312261069355399978262221877, 11.11829084628877236405602404271