| L(s) = 1 | + (1.58 + 0.620i)2-s + (0.433 + 1.67i)3-s + (0.647 + 0.601i)4-s + (0.761 + 0.519i)5-s + (−0.354 + 2.91i)6-s + (−0.654 − 2.56i)7-s + (−0.822 − 1.70i)8-s + (−2.62 + 1.45i)9-s + (0.881 + 1.29i)10-s + (−0.0278 − 0.184i)11-s + (−0.727 + 1.34i)12-s + (−2.33 + 1.86i)13-s + (0.556 − 4.45i)14-s + (−0.540 + 1.50i)15-s + (−0.372 − 4.97i)16-s + (3.71 + 1.14i)17-s + ⋯ |
| L(s) = 1 | + (1.11 + 0.438i)2-s + (0.250 + 0.968i)3-s + (0.323 + 0.300i)4-s + (0.340 + 0.232i)5-s + (−0.144 + 1.19i)6-s + (−0.247 − 0.968i)7-s + (−0.290 − 0.603i)8-s + (−0.874 + 0.484i)9-s + (0.278 + 0.408i)10-s + (−0.00838 − 0.0556i)11-s + (−0.209 + 0.388i)12-s + (−0.648 + 0.517i)13-s + (0.148 − 1.19i)14-s + (−0.139 + 0.387i)15-s + (−0.0931 − 1.24i)16-s + (0.901 + 0.278i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.514 - 0.857i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.514 - 0.857i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.63459 + 0.925765i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.63459 + 0.925765i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-0.433 - 1.67i)T \) |
| 7 | \( 1 + (0.654 + 2.56i)T \) |
| good | 2 | \( 1 + (-1.58 - 0.620i)T + (1.46 + 1.36i)T^{2} \) |
| 5 | \( 1 + (-0.761 - 0.519i)T + (1.82 + 4.65i)T^{2} \) |
| 11 | \( 1 + (0.0278 + 0.184i)T + (-10.5 + 3.24i)T^{2} \) |
| 13 | \( 1 + (2.33 - 1.86i)T + (2.89 - 12.6i)T^{2} \) |
| 17 | \( 1 + (-3.71 - 1.14i)T + (14.0 + 9.57i)T^{2} \) |
| 19 | \( 1 + (-2.14 + 1.24i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-0.363 - 1.17i)T + (-19.0 + 12.9i)T^{2} \) |
| 29 | \( 1 + (9.18 - 2.09i)T + (26.1 - 12.5i)T^{2} \) |
| 31 | \( 1 + (-6.62 - 3.82i)T + (15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (-2.37 + 2.20i)T + (2.76 - 36.8i)T^{2} \) |
| 41 | \( 1 + (5.40 - 2.60i)T + (25.5 - 32.0i)T^{2} \) |
| 43 | \( 1 + (-4.29 - 2.06i)T + (26.8 + 33.6i)T^{2} \) |
| 47 | \( 1 + (3.79 - 9.66i)T + (-34.4 - 31.9i)T^{2} \) |
| 53 | \( 1 + (1.45 - 1.56i)T + (-3.96 - 52.8i)T^{2} \) |
| 59 | \( 1 + (-8.18 + 5.58i)T + (21.5 - 54.9i)T^{2} \) |
| 61 | \( 1 + (3.96 + 4.27i)T + (-4.55 + 60.8i)T^{2} \) |
| 67 | \( 1 + (-5.08 + 8.81i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (0.367 + 0.0838i)T + (63.9 + 30.8i)T^{2} \) |
| 73 | \( 1 + (7.84 - 3.08i)T + (53.5 - 49.6i)T^{2} \) |
| 79 | \( 1 + (-7.67 - 13.2i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (0.837 - 1.04i)T + (-18.4 - 80.9i)T^{2} \) |
| 89 | \( 1 + (-5.96 - 0.899i)T + (85.0 + 26.2i)T^{2} \) |
| 97 | \( 1 + 5.60iT - 97T^{2} \) |
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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
−13.64907019204422355414132255758, −12.50593905961764996709913624314, −11.20870690103644283038350475624, −10.00755739341221699766675500155, −9.474023773137531805851001397563, −7.72818248218421579730700029017, −6.47419162265056592883897678416, −5.27693865190300198193174730479, −4.25412162216836010876040076763, −3.19253298710051704439491001567,
2.20764313582445215297493365859, 3.34790411110413128544472933226, 5.28340042034895223060041009052, 5.93704481712643768470779217352, 7.54200176723334283656913741606, 8.664568245988073281066159616862, 9.788936122620001252935470007915, 11.62039339060623483297274133215, 12.05950033500824562908945551550, 13.00310413419353883274116252637
