| L(s) = 1 | + (−4.78 + 3.01i)2-s + (13.9 − 6.95i)3-s + (13.8 − 28.8i)4-s + (−0.143 + 0.0826i)5-s + (−45.8 + 75.3i)6-s + (−192. − 111. i)7-s + (20.7 + 179. i)8-s + (146. − 194. i)9-s + (0.436 − 0.826i)10-s + (329. − 570. i)11-s + (−7.72 − 498. i)12-s + (43.8 + 76.0i)13-s + (1.25e3 − 48.1i)14-s + (−1.42 + 2.14i)15-s + (−641. − 798. i)16-s − 85.1i·17-s + ⋯ |
| L(s) = 1 | + (−0.846 + 0.532i)2-s + (0.894 − 0.446i)3-s + (0.432 − 0.901i)4-s + (−0.00256 + 0.00147i)5-s + (−0.519 + 0.854i)6-s + (−1.48 − 0.857i)7-s + (0.114 + 0.993i)8-s + (0.601 − 0.798i)9-s + (0.00137 − 0.00261i)10-s + (0.821 − 1.42i)11-s + (−0.0154 − 0.999i)12-s + (0.0720 + 0.124i)13-s + (1.71 − 0.0655i)14-s + (−0.00163 + 0.00246i)15-s + (−0.626 − 0.779i)16-s − 0.0714i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 36 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.298 + 0.954i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 36 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.298 + 0.954i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.912565 - 0.670425i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.912565 - 0.670425i\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (4.78 - 3.01i)T \) |
| 3 | \( 1 + (-13.9 + 6.95i)T \) |
| good | 5 | \( 1 + (0.143 - 0.0826i)T + (1.56e3 - 2.70e3i)T^{2} \) |
| 7 | \( 1 + (192. + 111. i)T + (8.40e3 + 1.45e4i)T^{2} \) |
| 11 | \( 1 + (-329. + 570. i)T + (-8.05e4 - 1.39e5i)T^{2} \) |
| 13 | \( 1 + (-43.8 - 76.0i)T + (-1.85e5 + 3.21e5i)T^{2} \) |
| 17 | \( 1 + 85.1iT - 1.41e6T^{2} \) |
| 19 | \( 1 + 2.30e3iT - 2.47e6T^{2} \) |
| 23 | \( 1 + (-1.32e3 - 2.29e3i)T + (-3.21e6 + 5.57e6i)T^{2} \) |
| 29 | \( 1 + (-1.33e3 - 772. i)T + (1.02e7 + 1.77e7i)T^{2} \) |
| 31 | \( 1 + (1.20e3 - 694. i)T + (1.43e7 - 2.47e7i)T^{2} \) |
| 37 | \( 1 - 4.40e3T + 6.93e7T^{2} \) |
| 41 | \( 1 + (3.14e3 - 1.81e3i)T + (5.79e7 - 1.00e8i)T^{2} \) |
| 43 | \( 1 + (2.42e3 + 1.39e3i)T + (7.35e7 + 1.27e8i)T^{2} \) |
| 47 | \( 1 + (4.97e3 - 8.61e3i)T + (-1.14e8 - 1.98e8i)T^{2} \) |
| 53 | \( 1 - 1.51e4iT - 4.18e8T^{2} \) |
| 59 | \( 1 + (-1.72e4 - 2.98e4i)T + (-3.57e8 + 6.19e8i)T^{2} \) |
| 61 | \( 1 + (-2.21e4 + 3.84e4i)T + (-4.22e8 - 7.31e8i)T^{2} \) |
| 67 | \( 1 + (-3.03e4 + 1.75e4i)T + (6.75e8 - 1.16e9i)T^{2} \) |
| 71 | \( 1 - 6.00e3T + 1.80e9T^{2} \) |
| 73 | \( 1 - 3.29e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + (2.39e4 + 1.38e4i)T + (1.53e9 + 2.66e9i)T^{2} \) |
| 83 | \( 1 + (-4.91e4 + 8.50e4i)T + (-1.96e9 - 3.41e9i)T^{2} \) |
| 89 | \( 1 + 1.91e4iT - 5.58e9T^{2} \) |
| 97 | \( 1 + (9.69e3 - 1.67e4i)T + (-4.29e9 - 7.43e9i)T^{2} \) |
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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
−15.41062563761536868391726753209, −13.96663372389640098143201479824, −13.24577062374519891164653216373, −11.23808871006587942339634836057, −9.649144806790493077129372114351, −8.873597176912375257241239448827, −7.28711509544721625573402958097, −6.35619260750913618647926889546, −3.29860540864970701159863532552, −0.802451424694785225069418199221,
2.28252895882951933323790400295, 3.80237092172780309742370744591, 6.72656917585115608088622779869, 8.366156491578286859981491643112, 9.567297455242403249449690203869, 10.09427677570815436017696304024, 12.14070350101021853069970202030, 12.92373751829204175033597850795, 14.73308284342252582243285105766, 15.82507011176499734724542028484
