L(s) = 1 | + (−4.10 + 7.10i)2-s + (−4.5 − 7.79i)3-s + (−17.6 − 30.6i)4-s + (−14.6 + 25.3i)5-s + 73.8·6-s + 27.7·8-s + (−40.5 + 70.1i)9-s + (−119. − 207. i)10-s + (−188. − 326. i)11-s + (−159. + 275. i)12-s + 509.·13-s + 263.·15-s + (452. − 783. i)16-s + (−802. − 1.39e3i)17-s + (−332. − 575. i)18-s + (−1.07e3 + 1.86e3i)19-s + ⋯ |
L(s) = 1 | + (−0.725 + 1.25i)2-s + (−0.288 − 0.499i)3-s + (−0.552 − 0.957i)4-s + (−0.261 + 0.452i)5-s + 0.837·6-s + 0.153·8-s + (−0.166 + 0.288i)9-s + (−0.379 − 0.657i)10-s + (−0.470 − 0.814i)11-s + (−0.319 + 0.552i)12-s + 0.836·13-s + 0.301·15-s + (0.441 − 0.764i)16-s + (−0.673 − 1.16i)17-s + (−0.241 − 0.418i)18-s + (−0.685 + 1.18i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0725 - 0.997i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.0725 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(3)\) |
\(\approx\) |
\(0.8597057629\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8597057629\) |
\(L(\frac{7}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (4.5 + 7.79i)T \) |
| 7 | \( 1 \) |
good | 2 | \( 1 + (4.10 - 7.10i)T + (-16 - 27.7i)T^{2} \) |
| 5 | \( 1 + (14.6 - 25.3i)T + (-1.56e3 - 2.70e3i)T^{2} \) |
| 11 | \( 1 + (188. + 326. i)T + (-8.05e4 + 1.39e5i)T^{2} \) |
| 13 | \( 1 - 509.T + 3.71e5T^{2} \) |
| 17 | \( 1 + (802. + 1.39e3i)T + (-7.09e5 + 1.22e6i)T^{2} \) |
| 19 | \( 1 + (1.07e3 - 1.86e3i)T + (-1.23e6 - 2.14e6i)T^{2} \) |
| 23 | \( 1 + (-2.21e3 + 3.84e3i)T + (-3.21e6 - 5.57e6i)T^{2} \) |
| 29 | \( 1 - 4.77e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + (-3.57e3 - 6.19e3i)T + (-1.43e7 + 2.47e7i)T^{2} \) |
| 37 | \( 1 + (2.78e3 - 4.82e3i)T + (-3.46e7 - 6.00e7i)T^{2} \) |
| 41 | \( 1 + 9.91e3T + 1.15e8T^{2} \) |
| 43 | \( 1 + 4.88e3T + 1.47e8T^{2} \) |
| 47 | \( 1 + (4.63e3 - 8.02e3i)T + (-1.14e8 - 1.98e8i)T^{2} \) |
| 53 | \( 1 + (-4.62e3 - 8.00e3i)T + (-2.09e8 + 3.62e8i)T^{2} \) |
| 59 | \( 1 + (-7.06e3 - 1.22e4i)T + (-3.57e8 + 6.19e8i)T^{2} \) |
| 61 | \( 1 + (-9.01e3 + 1.56e4i)T + (-4.22e8 - 7.31e8i)T^{2} \) |
| 67 | \( 1 + (-2.32e4 - 4.02e4i)T + (-6.75e8 + 1.16e9i)T^{2} \) |
| 71 | \( 1 - 6.42e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + (-3.04e4 - 5.27e4i)T + (-1.03e9 + 1.79e9i)T^{2} \) |
| 79 | \( 1 + (-3.57e4 + 6.18e4i)T + (-1.53e9 - 2.66e9i)T^{2} \) |
| 83 | \( 1 - 1.15e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + (3.91e4 - 6.77e4i)T + (-2.79e9 - 4.83e9i)T^{2} \) |
| 97 | \( 1 - 1.51e5T + 8.58e9T^{2} \) |
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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
−12.39817016108677389637744834931, −11.21914005913628593776357777607, −10.28232898905529654380140133676, −8.697325635188755279447497199518, −8.225222745872243650081194479335, −6.89586639485439963404188483161, −6.40015504109590319696457080618, −5.09075672067411186379103671752, −3.00076554229018547765915176319, −0.75247912593173601622975537789,
0.62610657226854368535548144900, 2.11316279803754402268277645691, 3.64786187903663186704431163738, 4.84981232753615582588047627124, 6.47701130014100605914325521206, 8.250938178089046788364976937750, 9.019400622042319330119384183452, 10.04482204968888367988345029136, 10.86590785171052728200747469509, 11.58322781690039135467838316792