L(s) = 1 | + (1.65 − 1.12i)2-s + (1.45 − 3.72i)4-s + (1.80 + 3.12i)5-s + (5.40 + 3.12i)7-s + (−1.80 − 7.79i)8-s + (6.5 + 3.12i)10-s + (10.5 + 6.06i)11-s + (8 + 13.8i)13-s + (12.4 − 0.945i)14-s + (−11.7 − 10.8i)16-s − 14.4·17-s − 37.4i·19-s + (14.2 − 2.17i)20-s + (24.1 − 1.83i)22-s + (15 − 8.66i)23-s + ⋯ |
L(s) = 1 | + (0.825 − 0.564i)2-s + (0.363 − 0.931i)4-s + (0.360 + 0.624i)5-s + (0.772 + 0.446i)7-s + (−0.225 − 0.974i)8-s + (0.650 + 0.312i)10-s + (0.954 + 0.551i)11-s + (0.615 + 1.06i)13-s + (0.889 − 0.0675i)14-s + (−0.735 − 0.677i)16-s − 0.848·17-s − 1.97i·19-s + (0.712 − 0.108i)20-s + (1.09 − 0.0834i)22-s + (0.652 − 0.376i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.854 + 0.519i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.854 + 0.519i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(2.97165 - 0.832995i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.97165 - 0.832995i\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-1.65 + 1.12i)T \) |
| 3 | \( 1 \) |
good | 5 | \( 1 + (-1.80 - 3.12i)T + (-12.5 + 21.6i)T^{2} \) |
| 7 | \( 1 + (-5.40 - 3.12i)T + (24.5 + 42.4i)T^{2} \) |
| 11 | \( 1 + (-10.5 - 6.06i)T + (60.5 + 104. i)T^{2} \) |
| 13 | \( 1 + (-8 - 13.8i)T + (-84.5 + 146. i)T^{2} \) |
| 17 | \( 1 + 14.4T + 289T^{2} \) |
| 19 | \( 1 + 37.4iT - 361T^{2} \) |
| 23 | \( 1 + (-15 + 8.66i)T + (264.5 - 458. i)T^{2} \) |
| 29 | \( 1 + (25.2 - 43.7i)T + (-420.5 - 728. i)T^{2} \) |
| 31 | \( 1 + (-5.40 + 3.12i)T + (480.5 - 832. i)T^{2} \) |
| 37 | \( 1 - 26T + 1.36e3T^{2} \) |
| 41 | \( 1 + (3.60 + 6.24i)T + (-840.5 + 1.45e3i)T^{2} \) |
| 43 | \( 1 + (10.8 + 6.24i)T + (924.5 + 1.60e3i)T^{2} \) |
| 47 | \( 1 + (3 + 1.73i)T + (1.10e3 + 1.91e3i)T^{2} \) |
| 53 | \( 1 + 68.5T + 2.80e3T^{2} \) |
| 59 | \( 1 + (66 - 38.1i)T + (1.74e3 - 3.01e3i)T^{2} \) |
| 61 | \( 1 + (4 - 6.92i)T + (-1.86e3 - 3.22e3i)T^{2} \) |
| 67 | \( 1 + (-54.0 + 31.2i)T + (2.24e3 - 3.88e3i)T^{2} \) |
| 71 | \( 1 - 62.3iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 19T + 5.32e3T^{2} \) |
| 79 | \( 1 + (43.2 + 24.9i)T + (3.12e3 + 5.40e3i)T^{2} \) |
| 83 | \( 1 + (-100.5 - 58.0i)T + (3.44e3 + 5.96e3i)T^{2} \) |
| 89 | \( 1 + 79.3T + 7.92e3T^{2} \) |
| 97 | \( 1 + (59.5 - 103. i)T + (-4.70e3 - 8.14e3i)T^{2} \) |
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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
−11.12186760756037032277247638772, −10.99393744203699362688407568837, −9.437690511329355316447645598298, −8.870599724525398338702100027390, −6.91091513569184527577375339600, −6.53063401222813232225393912394, −5.05413822200218538628551723749, −4.23228934585408209731725913604, −2.71616051575972780400563139406, −1.61639310371297592406930380257,
1.51518147768868096056671452768, 3.45196995980770222897671860214, 4.44532315316115764936499841648, 5.59314603648026225438929384515, 6.31141352011600886705035688831, 7.74254028958920831688388840358, 8.327112974094219319146669785282, 9.430358345283875385303548443314, 10.88260356308604109249757172805, 11.54456573042328334478734291307