| L(s) = 1 | + (−1.11 − 0.507i)2-s + (1.18 − 4.02i)3-s + (−1.64 − 1.89i)4-s + (−1.66 + 4.71i)5-s + (−3.35 + 3.87i)6-s + (1.89 − 13.1i)7-s + (2.23 + 7.62i)8-s + (−7.23 − 4.65i)9-s + (4.24 − 4.39i)10-s + (−11.1 + 5.08i)11-s + (−9.56 + 4.37i)12-s + (−6.17 + 0.887i)13-s + (−8.78 + 13.6i)14-s + (17.0 + 12.2i)15-s + (−0.0452 + 0.314i)16-s + (−2.88 + 3.32i)17-s + ⋯ |
| L(s) = 1 | + (−0.555 − 0.253i)2-s + (0.394 − 1.34i)3-s + (−0.410 − 0.473i)4-s + (−0.332 + 0.943i)5-s + (−0.559 + 0.645i)6-s + (0.270 − 1.88i)7-s + (0.279 + 0.953i)8-s + (−0.804 − 0.516i)9-s + (0.424 − 0.439i)10-s + (−1.01 + 0.462i)11-s + (−0.797 + 0.364i)12-s + (−0.474 + 0.0682i)13-s + (−0.627 + 0.976i)14-s + (1.13 + 0.817i)15-s + (−0.00282 + 0.0196i)16-s + (−0.169 + 0.195i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 115 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.986 + 0.162i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 115 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.986 + 0.162i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.0624888 - 0.764651i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0624888 - 0.764651i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + (1.66 - 4.71i)T \) |
| 23 | \( 1 + (21.6 + 7.70i)T \) |
| good | 2 | \( 1 + (1.11 + 0.507i)T + (2.61 + 3.02i)T^{2} \) |
| 3 | \( 1 + (-1.18 + 4.02i)T + (-7.57 - 4.86i)T^{2} \) |
| 7 | \( 1 + (-1.89 + 13.1i)T + (-47.0 - 13.8i)T^{2} \) |
| 11 | \( 1 + (11.1 - 5.08i)T + (79.2 - 91.4i)T^{2} \) |
| 13 | \( 1 + (6.17 - 0.887i)T + (162. - 47.6i)T^{2} \) |
| 17 | \( 1 + (2.88 - 3.32i)T + (-41.1 - 286. i)T^{2} \) |
| 19 | \( 1 + (-12.6 + 10.9i)T + (51.3 - 357. i)T^{2} \) |
| 29 | \( 1 + (-29.7 + 34.3i)T + (-119. - 832. i)T^{2} \) |
| 31 | \( 1 + (-36.6 + 10.7i)T + (808. - 519. i)T^{2} \) |
| 37 | \( 1 + (13.3 + 8.57i)T + (568. + 1.24e3i)T^{2} \) |
| 41 | \( 1 + (-5.27 + 3.38i)T + (698. - 1.52e3i)T^{2} \) |
| 43 | \( 1 + (-51.6 - 15.1i)T + (1.55e3 + 9.99e2i)T^{2} \) |
| 47 | \( 1 - 11.0iT - 2.20e3T^{2} \) |
| 53 | \( 1 + (-2.03 + 14.1i)T + (-2.69e3 - 791. i)T^{2} \) |
| 59 | \( 1 + (12.7 + 88.8i)T + (-3.33e3 + 980. i)T^{2} \) |
| 61 | \( 1 + (-17.1 - 58.5i)T + (-3.13e3 + 2.01e3i)T^{2} \) |
| 67 | \( 1 + (-13.2 + 29.1i)T + (-2.93e3 - 3.39e3i)T^{2} \) |
| 71 | \( 1 + (23.6 - 51.8i)T + (-3.30e3 - 3.80e3i)T^{2} \) |
| 73 | \( 1 + (15.8 - 13.6i)T + (758. - 5.27e3i)T^{2} \) |
| 79 | \( 1 + (-16.7 + 2.40i)T + (5.98e3 - 1.75e3i)T^{2} \) |
| 83 | \( 1 + (-32.0 - 20.5i)T + (2.86e3 + 6.26e3i)T^{2} \) |
| 89 | \( 1 + (0.945 - 3.21i)T + (-6.66e3 - 4.28e3i)T^{2} \) |
| 97 | \( 1 + (-96.1 + 61.7i)T + (3.90e3 - 8.55e3i)T^{2} \) |
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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
−13.12145837296835289538830852779, −11.65381924326123126768169854037, −10.52447713075830645862467261750, −9.967724828717782071152271154873, −7.989250554684175674865662743131, −7.61967116249029967240482182321, −6.54085946548751913362713076195, −4.45691318214680782147537155129, −2.37748788828555765186644526546, −0.64129588789467338659403886228,
3.06167215618501156314115177870, 4.62473130632502649903936289023, 5.48314105002852990285520764428, 7.953107784543381217446328779727, 8.645875210774136433352615174409, 9.274105484140122324037695351864, 10.23067139877609992963915012363, 11.90797552566109288311235242839, 12.55405793604937859562488706275, 13.95081794019830566627351467290
