| L(s) = 1 | + (1.98 − 0.275i)2-s + (3.84 − 1.09i)4-s + (0.425 − 0.154i)5-s + (6.84 − 1.20i)7-s + (7.32 − 3.22i)8-s + (0.800 − 0.424i)10-s + (−0.541 + 1.48i)11-s + (−5.56 − 4.66i)13-s + (13.2 − 4.27i)14-s + (13.6 − 8.41i)16-s + (−8.72 + 15.1i)17-s + (24.4 − 14.1i)19-s + (1.46 − 1.06i)20-s + (−0.662 + 3.09i)22-s + (−5.82 − 1.02i)23-s + ⋯ |
| L(s) = 1 | + (0.990 − 0.137i)2-s + (0.961 − 0.273i)4-s + (0.0851 − 0.0309i)5-s + (0.977 − 0.172i)7-s + (0.915 − 0.403i)8-s + (0.0800 − 0.0424i)10-s + (−0.0492 + 0.135i)11-s + (−0.427 − 0.358i)13-s + (0.944 − 0.305i)14-s + (0.850 − 0.525i)16-s + (−0.513 + 0.888i)17-s + (1.28 − 0.743i)19-s + (0.0734 − 0.0530i)20-s + (−0.0301 + 0.140i)22-s + (−0.253 − 0.0446i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.916 + 0.401i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.916 + 0.401i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(3.32407 - 0.695938i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.32407 - 0.695938i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-1.98 + 0.275i)T \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 + (-0.425 + 0.154i)T + (19.1 - 16.0i)T^{2} \) |
| 7 | \( 1 + (-6.84 + 1.20i)T + (46.0 - 16.7i)T^{2} \) |
| 11 | \( 1 + (0.541 - 1.48i)T + (-92.6 - 77.7i)T^{2} \) |
| 13 | \( 1 + (5.56 + 4.66i)T + (29.3 + 166. i)T^{2} \) |
| 17 | \( 1 + (8.72 - 15.1i)T + (-144.5 - 250. i)T^{2} \) |
| 19 | \( 1 + (-24.4 + 14.1i)T + (180.5 - 312. i)T^{2} \) |
| 23 | \( 1 + (5.82 + 1.02i)T + (497. + 180. i)T^{2} \) |
| 29 | \( 1 + (-34.2 + 28.7i)T + (146. - 828. i)T^{2} \) |
| 31 | \( 1 + (-19.5 - 3.45i)T + (903. + 328. i)T^{2} \) |
| 37 | \( 1 + (13.6 - 23.6i)T + (-684.5 - 1.18e3i)T^{2} \) |
| 41 | \( 1 + (28.7 + 24.1i)T + (291. + 1.65e3i)T^{2} \) |
| 43 | \( 1 + (12.2 - 33.5i)T + (-1.41e3 - 1.18e3i)T^{2} \) |
| 47 | \( 1 + (13.1 - 2.31i)T + (2.07e3 - 755. i)T^{2} \) |
| 53 | \( 1 + 87.7T + 2.80e3T^{2} \) |
| 59 | \( 1 + (-0.702 - 1.92i)T + (-2.66e3 + 2.23e3i)T^{2} \) |
| 61 | \( 1 + (-15.9 - 90.5i)T + (-3.49e3 + 1.27e3i)T^{2} \) |
| 67 | \( 1 + (68.8 - 82.0i)T + (-779. - 4.42e3i)T^{2} \) |
| 71 | \( 1 + (91.1 + 52.6i)T + (2.52e3 + 4.36e3i)T^{2} \) |
| 73 | \( 1 + (57.3 + 99.4i)T + (-2.66e3 + 4.61e3i)T^{2} \) |
| 79 | \( 1 + (-36.2 - 43.1i)T + (-1.08e3 + 6.14e3i)T^{2} \) |
| 83 | \( 1 + (-104. - 124. i)T + (-1.19e3 + 6.78e3i)T^{2} \) |
| 89 | \( 1 + (15.6 + 27.0i)T + (-3.96e3 + 6.85e3i)T^{2} \) |
| 97 | \( 1 + (7.06 + 2.57i)T + (7.20e3 + 6.04e3i)T^{2} \) |
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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
−11.56855591031922506099428344127, −10.62771471966040634894482894764, −9.754056632625217934246039991744, −8.239008993851109900259890167986, −7.40855555556994765940158197674, −6.26005325734669652211950417213, −5.13802245436277302162631323771, −4.36121280006157679066165008949, −2.92999497857376260652017468259, −1.52206712202674785637289332638,
1.77426583932599299429204770986, 3.13276935086511065433931027196, 4.57133908926137795141320904554, 5.27301801665567925239527558757, 6.46704335145894223825607650529, 7.51109536009607762298147314466, 8.347427911461831126403234914640, 9.737240294026678857022814568231, 10.80804486451326925339444725557, 11.81021456355450564145894405168
