| L(s) = 1 | + (1.80 + 2.26i)2-s + (−3.86 + 4.85i)3-s + (−0.0889 + 0.389i)4-s + (−19.3 + 9.33i)5-s − 17.9·6-s + 25.5·7-s + (19.8 − 9.55i)8-s + (−2.56 − 11.2i)9-s + (−56.1 − 27.0i)10-s + (9.85 + 43.1i)11-s + (−1.54 − 1.94i)12-s + (−20.3 + 9.79i)13-s + (46.2 + 57.9i)14-s + (29.6 − 130. i)15-s + (60.4 + 29.0i)16-s + (28.8 + 13.8i)17-s + ⋯ |
| L(s) = 1 | + (0.638 + 0.801i)2-s + (−0.744 + 0.933i)3-s + (−0.0111 + 0.0487i)4-s + (−1.73 + 0.834i)5-s − 1.22·6-s + 1.38·7-s + (0.877 − 0.422i)8-s + (−0.0948 − 0.415i)9-s + (−1.77 − 0.855i)10-s + (0.270 + 1.18i)11-s + (−0.0372 − 0.0466i)12-s + (−0.434 + 0.209i)13-s + (0.882 + 1.10i)14-s + (0.511 − 2.23i)15-s + (0.943 + 0.454i)16-s + (0.411 + 0.198i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.769 - 0.638i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.769 - 0.638i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.429089 + 1.18938i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.429089 + 1.18938i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 43 | \( 1 + (241. + 145. i)T \) |
| good | 2 | \( 1 + (-1.80 - 2.26i)T + (-1.78 + 7.79i)T^{2} \) |
| 3 | \( 1 + (3.86 - 4.85i)T + (-6.00 - 26.3i)T^{2} \) |
| 5 | \( 1 + (19.3 - 9.33i)T + (77.9 - 97.7i)T^{2} \) |
| 7 | \( 1 - 25.5T + 343T^{2} \) |
| 11 | \( 1 + (-9.85 - 43.1i)T + (-1.19e3 + 577. i)T^{2} \) |
| 13 | \( 1 + (20.3 - 9.79i)T + (1.36e3 - 1.71e3i)T^{2} \) |
| 17 | \( 1 + (-28.8 - 13.8i)T + (3.06e3 + 3.84e3i)T^{2} \) |
| 19 | \( 1 + (-17.0 + 74.7i)T + (-6.17e3 - 2.97e3i)T^{2} \) |
| 23 | \( 1 + (11.2 + 49.1i)T + (-1.09e4 + 5.27e3i)T^{2} \) |
| 29 | \( 1 + (32.4 + 40.6i)T + (-5.42e3 + 2.37e4i)T^{2} \) |
| 31 | \( 1 + (-167. - 209. i)T + (-6.62e3 + 2.90e4i)T^{2} \) |
| 37 | \( 1 - 73.1T + 5.06e4T^{2} \) |
| 41 | \( 1 + (-80.0 - 100. i)T + (-1.53e4 + 6.71e4i)T^{2} \) |
| 47 | \( 1 + (35.1 - 154. i)T + (-9.35e4 - 4.50e4i)T^{2} \) |
| 53 | \( 1 + (-377. - 181. i)T + (9.28e4 + 1.16e5i)T^{2} \) |
| 59 | \( 1 + (188. + 90.5i)T + (1.28e5 + 1.60e5i)T^{2} \) |
| 61 | \( 1 + (-191. + 240. i)T + (-5.05e4 - 2.21e5i)T^{2} \) |
| 67 | \( 1 + (76.7 - 336. i)T + (-2.70e5 - 1.30e5i)T^{2} \) |
| 71 | \( 1 + (-88.8 + 389. i)T + (-3.22e5 - 1.55e5i)T^{2} \) |
| 73 | \( 1 + (-79.0 + 38.0i)T + (2.42e5 - 3.04e5i)T^{2} \) |
| 79 | \( 1 - 221.T + 4.93e5T^{2} \) |
| 83 | \( 1 + (-270. + 339. i)T + (-1.27e5 - 5.57e5i)T^{2} \) |
| 89 | \( 1 + (11.6 - 14.6i)T + (-1.56e5 - 6.87e5i)T^{2} \) |
| 97 | \( 1 + (74.3 + 325. i)T + (-8.22e5 + 3.95e5i)T^{2} \) |
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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
−15.51619443713689443102753735608, −15.07773711687839441623589598350, −14.29646992550755877584271436067, −12.06740110097035783919159731403, −11.18961417539017966380485098565, −10.28618886042847066597425766609, −7.895628370387353746715058817928, −6.91978306564318152384787917642, −4.88204960621895701418285596247, −4.30872012465056082054744340030,
1.06020838365574507287228057921, 3.85155540531044645993460046598, 5.23411576592030000985164828699, 7.61884116125702875910305488037, 8.217741381115197471544225475030, 11.15699122883051725867263134591, 11.73452094515450525686824792869, 12.19874482697832030679273710432, 13.39325764828228457274173231356, 14.81422014057064467659740324968
