| L(s) = 1 | + (−0.826 − 0.954i)2-s + (−13.1 − 8.42i)3-s + (8.88 − 61.7i)4-s + (46.0 + 21.0i)5-s + (2.80 + 19.4i)6-s + (134. − 458. i)7-s + (−134. + 86.2i)8-s + (100. + 221. i)9-s + (−18.0 − 61.3i)10-s + (−54.3 − 47.1i)11-s + (−637. + 735. i)12-s + (3.57e3 − 1.04e3i)13-s + (−548. + 250. i)14-s + (−426. − 663. i)15-s + (−3.63e3 − 1.06e3i)16-s + (−6.29e3 + 905. i)17-s + ⋯ |
| L(s) = 1 | + (−0.103 − 0.119i)2-s + (−0.485 − 0.312i)3-s + (0.138 − 0.965i)4-s + (0.368 + 0.168i)5-s + (0.0129 + 0.0901i)6-s + (0.392 − 1.33i)7-s + (−0.262 + 0.168i)8-s + (0.138 + 0.303i)9-s + (−0.0180 − 0.0613i)10-s + (−0.0408 − 0.0354i)11-s + (−0.368 + 0.425i)12-s + (1.62 − 0.477i)13-s + (−0.199 + 0.0913i)14-s + (−0.126 − 0.196i)15-s + (−0.888 − 0.260i)16-s + (−1.28 + 0.184i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 69 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.936 + 0.350i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 69 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (-0.936 + 0.350i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(0.225090 - 1.24524i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.225090 - 1.24524i\) |
| \(L(4)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (13.1 + 8.42i)T \) |
| 23 | \( 1 + (1.08e4 - 5.56e3i)T \) |
| good | 2 | \( 1 + (0.826 + 0.954i)T + (-9.10 + 63.3i)T^{2} \) |
| 5 | \( 1 + (-46.0 - 21.0i)T + (1.02e4 + 1.18e4i)T^{2} \) |
| 7 | \( 1 + (-134. + 458. i)T + (-9.89e4 - 6.36e4i)T^{2} \) |
| 11 | \( 1 + (54.3 + 47.1i)T + (2.52e5 + 1.75e6i)T^{2} \) |
| 13 | \( 1 + (-3.57e3 + 1.04e3i)T + (4.06e6 - 2.60e6i)T^{2} \) |
| 17 | \( 1 + (6.29e3 - 905. i)T + (2.31e7 - 6.80e6i)T^{2} \) |
| 19 | \( 1 + (1.06e3 + 153. i)T + (4.51e7 + 1.32e7i)T^{2} \) |
| 29 | \( 1 + (3.94e3 + 2.74e4i)T + (-5.70e8 + 1.67e8i)T^{2} \) |
| 31 | \( 1 + (-7.04e3 + 4.52e3i)T + (3.68e8 - 8.07e8i)T^{2} \) |
| 37 | \( 1 + (4.89e4 - 2.23e4i)T + (1.68e9 - 1.93e9i)T^{2} \) |
| 41 | \( 1 + (4.49e4 - 9.85e4i)T + (-3.11e9 - 3.58e9i)T^{2} \) |
| 43 | \( 1 + (-1.24e4 + 1.93e4i)T + (-2.62e9 - 5.75e9i)T^{2} \) |
| 47 | \( 1 - 1.11e5T + 1.07e10T^{2} \) |
| 53 | \( 1 + (-3.78e4 + 1.28e5i)T + (-1.86e10 - 1.19e10i)T^{2} \) |
| 59 | \( 1 + (-1.50e5 + 4.41e4i)T + (3.54e10 - 2.28e10i)T^{2} \) |
| 61 | \( 1 + (-1.62e5 - 2.52e5i)T + (-2.14e10 + 4.68e10i)T^{2} \) |
| 67 | \( 1 + (-1.34e5 + 1.16e5i)T + (1.28e10 - 8.95e10i)T^{2} \) |
| 71 | \( 1 + (-6.46e4 - 7.46e4i)T + (-1.82e10 + 1.26e11i)T^{2} \) |
| 73 | \( 1 + (-3.92e4 + 2.73e5i)T + (-1.45e11 - 4.26e10i)T^{2} \) |
| 79 | \( 1 + (2.27e5 + 7.75e5i)T + (-2.04e11 + 1.31e11i)T^{2} \) |
| 83 | \( 1 + (-4.88e5 + 2.22e5i)T + (2.14e11 - 2.47e11i)T^{2} \) |
| 89 | \( 1 + (6.69e5 - 1.04e6i)T + (-2.06e11 - 4.52e11i)T^{2} \) |
| 97 | \( 1 + (2.63e5 + 1.20e5i)T + (5.45e11 + 6.29e11i)T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−13.52597916047228695047657617839, −11.52236857165586538196834024103, −10.75504429905486551558168620080, −9.999580400859262730081696176634, −8.291115527000191493092254454669, −6.73377665074775074547669878299, −5.84240096792400376081674263202, −4.21775049586184777735227240282, −1.77822577634160957915626745211, −0.54061466723394142993225165910,
2.09621832207997201887692464387, 3.91513287968310722690647622222, 5.53701439995155571273858677702, 6.72157072851772489893774141848, 8.535508115349068862021969679137, 9.043502790704314201222361183857, 10.90346275231139188752278230506, 11.78100067679592448909085623977, 12.71864936262561282528033716852, 13.87429274216299727166589866048
