| L(s) = 1 | + (−5.51 − 0.621i)2-s + (67.5 + 42.4i)3-s + (−219. − 50.1i)4-s + (264. − 211. i)5-s + (−346. − 276. i)6-s + (606. + 2.65e3i)7-s + (2.52e3 + 882. i)8-s + (−83.1 − 172. i)9-s + (−1.59e3 + 1.00e3i)10-s + (−1.96e3 + 689. i)11-s + (−1.27e4 − 1.27e4i)12-s + (−1.45e4 + 3.02e4i)13-s + (−1.69e3 − 1.50e4i)14-s + (2.68e4 − 3.02e3i)15-s + (3.85e4 + 1.85e4i)16-s + (−8.33e4 + 8.33e4i)17-s + ⋯ |
| L(s) = 1 | + (−0.344 − 0.0388i)2-s + (0.834 + 0.524i)3-s + (−0.857 − 0.195i)4-s + (0.423 − 0.337i)5-s + (−0.267 − 0.213i)6-s + (0.252 + 1.10i)7-s + (0.615 + 0.215i)8-s + (−0.0126 − 0.0263i)9-s + (−0.159 + 0.100i)10-s + (−0.134 + 0.0470i)11-s + (−0.612 − 0.612i)12-s + (−0.510 + 1.05i)13-s + (−0.0440 − 0.391i)14-s + (0.530 − 0.0597i)15-s + (0.588 + 0.283i)16-s + (−0.997 + 0.997i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 29 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.218 - 0.975i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 29 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (-0.218 - 0.975i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{9}{2})\) |
\(\approx\) |
\(0.849947 + 1.06124i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.849947 + 1.06124i\) |
| \(L(5)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 29 | \( 1 + (-3.89e5 + 5.90e5i)T \) |
| good | 2 | \( 1 + (5.51 + 0.621i)T + (249. + 56.9i)T^{2} \) |
| 3 | \( 1 + (-67.5 - 42.4i)T + (2.84e3 + 5.91e3i)T^{2} \) |
| 5 | \( 1 + (-264. + 211. i)T + (8.69e4 - 3.80e5i)T^{2} \) |
| 7 | \( 1 + (-606. - 2.65e3i)T + (-5.19e6 + 2.50e6i)T^{2} \) |
| 11 | \( 1 + (1.96e3 - 689. i)T + (1.67e8 - 1.33e8i)T^{2} \) |
| 13 | \( 1 + (1.45e4 - 3.02e4i)T + (-5.08e8 - 6.37e8i)T^{2} \) |
| 17 | \( 1 + (8.33e4 - 8.33e4i)T - 6.97e9iT^{2} \) |
| 19 | \( 1 + (-9.73e4 - 1.54e5i)T + (-7.36e9 + 1.53e10i)T^{2} \) |
| 23 | \( 1 + (1.75e5 - 2.20e5i)T + (-1.74e10 - 7.63e10i)T^{2} \) |
| 31 | \( 1 + (4.07e5 + 4.59e4i)T + (8.31e11 + 1.89e11i)T^{2} \) |
| 37 | \( 1 + (-2.81e6 - 9.83e5i)T + (2.74e12 + 2.18e12i)T^{2} \) |
| 41 | \( 1 + (3.55e5 + 3.55e5i)T + 7.98e12iT^{2} \) |
| 43 | \( 1 + (4.57e5 + 4.06e6i)T + (-1.13e13 + 2.60e12i)T^{2} \) |
| 47 | \( 1 + (1.42e6 + 4.07e6i)T + (-1.86e13 + 1.48e13i)T^{2} \) |
| 53 | \( 1 + (-4.24e6 - 5.32e6i)T + (-1.38e13 + 6.06e13i)T^{2} \) |
| 59 | \( 1 - 4.96e6T + 1.46e14T^{2} \) |
| 61 | \( 1 + (4.06e6 + 2.55e6i)T + (8.31e13 + 1.72e14i)T^{2} \) |
| 67 | \( 1 + (-8.71e6 - 1.80e7i)T + (-2.53e14 + 3.17e14i)T^{2} \) |
| 71 | \( 1 + (-2.78e6 + 5.78e6i)T + (-4.02e14 - 5.04e14i)T^{2} \) |
| 73 | \( 1 + (-2.53e6 + 2.85e5i)T + (7.86e14 - 1.79e14i)T^{2} \) |
| 79 | \( 1 + (1.94e7 - 5.54e7i)T + (-1.18e15 - 9.45e14i)T^{2} \) |
| 83 | \( 1 + (-1.60e7 + 7.01e7i)T + (-2.02e15 - 9.77e14i)T^{2} \) |
| 89 | \( 1 + (8.67e5 + 9.77e4i)T + (3.83e15 + 8.75e14i)T^{2} \) |
| 97 | \( 1 + (4.90e7 - 3.08e7i)T + (3.40e15 - 7.06e15i)T^{2} \) |
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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
−15.39820873453559934521695637776, −14.47159737295585066885950004457, −13.43592061455616642613785336766, −11.90244296889944047704935312098, −9.888035566956992387836333374889, −9.149548963121640347956134245584, −8.242884084759795297261300249343, −5.66500572745299419140333294918, −4.07904927796382201480406924885, −1.93923243646766680365825923839,
0.62930571139063029669309569374, 2.75487151873753838591224133370, 4.73502181738732326823734731541, 7.19633953656456802210706892458, 8.165926027896870405730414922977, 9.538717316143418346763194304104, 10.79431090312225406793401683810, 12.97082611034209599496611292286, 13.72388952839550030729858018271, 14.47066566658706077785304138759
