| L(s) = 1 | + (1.03 − 3.86i)2-s + (−1.18 − 0.683i)3-s + (−13.8 − 7.99i)4-s + (−16.2 − 28.1i)5-s + (−3.86 + 3.86i)6-s + (47.8 + 10.4i)7-s + (−45.2 + 45.2i)8-s + (−39.5 − 68.5i)9-s + (−125. + 33.7i)10-s + (46.9 + 27.1i)11-s + (10.9 + 18.9i)12-s + 223.·13-s + (89.8 − 174. i)14-s + 44.4i·15-s + (128. + 221. i)16-s + (22.6 − 39.1i)17-s + ⋯ |
| L(s) = 1 | + (0.258 − 0.965i)2-s + (−0.131 − 0.0759i)3-s + (−0.866 − 0.499i)4-s + (−0.650 − 1.12i)5-s + (−0.107 + 0.107i)6-s + (0.977 + 0.213i)7-s + (−0.706 + 0.707i)8-s + (−0.488 − 0.846i)9-s + (−1.25 + 0.337i)10-s + (0.388 + 0.224i)11-s + (0.0759 + 0.131i)12-s + 1.32·13-s + (0.458 − 0.888i)14-s + 0.197i·15-s + (0.500 + 0.865i)16-s + (0.0782 − 0.135i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 28 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.718 + 0.695i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 28 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.718 + 0.695i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(0.475293 - 1.17377i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.475293 - 1.17377i\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-1.03 + 3.86i)T \) |
| 7 | \( 1 + (-47.8 - 10.4i)T \) |
| good | 3 | \( 1 + (1.18 + 0.683i)T + (40.5 + 70.1i)T^{2} \) |
| 5 | \( 1 + (16.2 + 28.1i)T + (-312.5 + 541. i)T^{2} \) |
| 11 | \( 1 + (-46.9 - 27.1i)T + (7.32e3 + 1.26e4i)T^{2} \) |
| 13 | \( 1 - 223.T + 2.85e4T^{2} \) |
| 17 | \( 1 + (-22.6 + 39.1i)T + (-4.17e4 - 7.23e4i)T^{2} \) |
| 19 | \( 1 + (-454. + 262. i)T + (6.51e4 - 1.12e5i)T^{2} \) |
| 23 | \( 1 + (640. - 369. i)T + (1.39e5 - 2.42e5i)T^{2} \) |
| 29 | \( 1 + 392.T + 7.07e5T^{2} \) |
| 31 | \( 1 + (494. + 285. i)T + (4.61e5 + 7.99e5i)T^{2} \) |
| 37 | \( 1 + (-162. - 281. i)T + (-9.37e5 + 1.62e6i)T^{2} \) |
| 41 | \( 1 - 110.T + 2.82e6T^{2} \) |
| 43 | \( 1 - 2.81e3iT - 3.41e6T^{2} \) |
| 47 | \( 1 + (-877. + 506. i)T + (2.43e6 - 4.22e6i)T^{2} \) |
| 53 | \( 1 + (-2.25e3 + 3.91e3i)T + (-3.94e6 - 6.83e6i)T^{2} \) |
| 59 | \( 1 + (-3.88e3 - 2.24e3i)T + (6.05e6 + 1.04e7i)T^{2} \) |
| 61 | \( 1 + (593. + 1.02e3i)T + (-6.92e6 + 1.19e7i)T^{2} \) |
| 67 | \( 1 + (-3.46e3 - 1.99e3i)T + (1.00e7 + 1.74e7i)T^{2} \) |
| 71 | \( 1 - 2.62e3iT - 2.54e7T^{2} \) |
| 73 | \( 1 + (254. - 440. i)T + (-1.41e7 - 2.45e7i)T^{2} \) |
| 79 | \( 1 + (8.41e3 - 4.85e3i)T + (1.94e7 - 3.37e7i)T^{2} \) |
| 83 | \( 1 + 7.48e3iT - 4.74e7T^{2} \) |
| 89 | \( 1 + (-2.12e3 - 3.68e3i)T + (-3.13e7 + 5.43e7i)T^{2} \) |
| 97 | \( 1 - 1.46e4T + 8.85e7T^{2} \) |
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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
−15.92572899224841232059475521824, −14.55769269101559188169415124561, −13.24443453523348587867593060907, −11.86079646192840736822131178396, −11.43807866269242762629347874608, −9.331282827185544967790229144096, −8.302270820021118393258812745303, −5.48287639683392677973300248135, −3.93522899306839967085984844302, −1.08204620260190674660965471130,
3.81079373562539591273600297221, 5.74735519923544617956442717348, 7.40727910069569561267254728000, 8.418434733376100291340500128380, 10.65733039557870036413904452088, 11.76931870932301300508859771204, 13.84187411074542671267844440172, 14.40614702169899765219310762810, 15.68394671856760574199058632483, 16.68780267760308978159540476949
