| L(s) = 1 | + (−6.02 + 3.87i)2-s + (−3.84 + 26.7i)3-s + (−31.8 + 69.7i)4-s + (331. − 97.4i)5-s + (−80.3 − 175. i)6-s + (−36.1 − 41.6i)7-s + (−208. − 1.45e3i)8-s + (−699. − 205. i)9-s + (−1.62e3 + 1.87e3i)10-s + (−3.24e3 − 2.08e3i)11-s + (−1.74e3 − 1.11e3i)12-s + (−4.80e3 + 5.55e3i)13-s + (378. + 111. i)14-s + (1.32e3 + 9.23e3i)15-s + (448. + 517. i)16-s + (−8.19e3 − 1.79e4i)17-s + ⋯ |
| L(s) = 1 | + (−0.532 + 0.342i)2-s + (−0.0821 + 0.571i)3-s + (−0.248 + 0.545i)4-s + (1.18 − 0.348i)5-s + (−0.151 − 0.332i)6-s + (−0.0397 − 0.0459i)7-s + (−0.144 − 1.00i)8-s + (−0.319 − 0.0939i)9-s + (−0.512 + 0.591i)10-s + (−0.734 − 0.471i)11-s + (−0.291 − 0.187i)12-s + (−0.607 + 0.700i)13-s + (0.0369 + 0.0108i)14-s + (0.101 + 0.706i)15-s + (0.0273 + 0.0316i)16-s + (−0.404 − 0.885i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 69 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.398 + 0.917i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 69 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (0.398 + 0.917i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(0.545295 - 0.357621i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.545295 - 0.357621i\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (3.84 - 26.7i)T \) |
| 23 | \( 1 + (9.57e3 + 5.75e4i)T \) |
| good | 2 | \( 1 + (6.02 - 3.87i)T + (53.1 - 116. i)T^{2} \) |
| 5 | \( 1 + (-331. + 97.4i)T + (6.57e4 - 4.22e4i)T^{2} \) |
| 7 | \( 1 + (36.1 + 41.6i)T + (-1.17e5 + 8.15e5i)T^{2} \) |
| 11 | \( 1 + (3.24e3 + 2.08e3i)T + (8.09e6 + 1.77e7i)T^{2} \) |
| 13 | \( 1 + (4.80e3 - 5.55e3i)T + (-8.93e6 - 6.21e7i)T^{2} \) |
| 17 | \( 1 + (8.19e3 + 1.79e4i)T + (-2.68e8 + 3.10e8i)T^{2} \) |
| 19 | \( 1 + (-206. + 452. i)T + (-5.85e8 - 6.75e8i)T^{2} \) |
| 29 | \( 1 + (3.12e4 + 6.83e4i)T + (-1.12e10 + 1.30e10i)T^{2} \) |
| 31 | \( 1 + (1.47e4 + 1.02e5i)T + (-2.63e10 + 7.75e9i)T^{2} \) |
| 37 | \( 1 + (1.47e5 + 4.32e4i)T + (7.98e10 + 5.13e10i)T^{2} \) |
| 41 | \( 1 + (-2.18e5 + 6.42e4i)T + (1.63e11 - 1.05e11i)T^{2} \) |
| 43 | \( 1 + (-6.90e4 + 4.80e5i)T + (-2.60e11 - 7.65e10i)T^{2} \) |
| 47 | \( 1 - 1.19e6T + 5.06e11T^{2} \) |
| 53 | \( 1 + (-3.07e5 - 3.54e5i)T + (-1.67e11 + 1.16e12i)T^{2} \) |
| 59 | \( 1 + (1.88e6 - 2.17e6i)T + (-3.54e11 - 2.46e12i)T^{2} \) |
| 61 | \( 1 + (8.54e4 + 5.94e5i)T + (-3.01e12 + 8.85e11i)T^{2} \) |
| 67 | \( 1 + (2.42e6 - 1.56e6i)T + (2.51e12 - 5.51e12i)T^{2} \) |
| 71 | \( 1 + (2.14e5 - 1.37e5i)T + (3.77e12 - 8.27e12i)T^{2} \) |
| 73 | \( 1 + (2.35e5 - 5.15e5i)T + (-7.23e12 - 8.34e12i)T^{2} \) |
| 79 | \( 1 + (-6.75e5 + 7.79e5i)T + (-2.73e12 - 1.90e13i)T^{2} \) |
| 83 | \( 1 + (5.72e6 + 1.68e6i)T + (2.28e13 + 1.46e13i)T^{2} \) |
| 89 | \( 1 + (-6.40e5 + 4.45e6i)T + (-4.24e13 - 1.24e13i)T^{2} \) |
| 97 | \( 1 + (-7.81e6 + 2.29e6i)T + (6.79e13 - 4.36e13i)T^{2} \) |
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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
−13.25789112807497530955619920408, −12.01510107684660404438863814373, −10.43900721893299098969977418022, −9.454831395734829677222857546543, −8.731052971483151042560730186488, −7.23775428624057591564902230103, −5.75484821004914132872154985915, −4.38324810580697303697662739860, −2.52610955469917625483665793678, −0.27133385897705577403688751140,
1.47421758894143323627898359124, 2.52259878424925628704329194450, 5.20522430772571919080735311321, 6.14843655490980672019430696089, 7.71234918160225002876397109235, 9.148573829090733116046286415228, 10.14839447744380538296131039391, 10.86765726070789814112812764532, 12.48740839947863081158649035568, 13.48398787271397910832431990560
