| L(s) = 1 | + (0.841 − 0.540i)2-s + (−0.142 + 0.989i)3-s + (0.415 − 0.909i)4-s + (−1.32 + 0.387i)5-s + (0.415 + 0.909i)6-s + (−0.654 − 0.755i)7-s + (−0.142 − 0.989i)8-s + (−0.959 − 0.281i)9-s + (−0.901 + 1.04i)10-s + (−2.01 − 1.29i)11-s + (0.841 + 0.540i)12-s + (−3.79 + 4.38i)13-s + (−0.959 − 0.281i)14-s + (−0.195 − 1.36i)15-s + (−0.654 − 0.755i)16-s + (1.41 + 3.09i)17-s + ⋯ |
| L(s) = 1 | + (0.594 − 0.382i)2-s + (−0.0821 + 0.571i)3-s + (0.207 − 0.454i)4-s + (−0.590 + 0.173i)5-s + (0.169 + 0.371i)6-s + (−0.247 − 0.285i)7-s + (−0.0503 − 0.349i)8-s + (−0.319 − 0.0939i)9-s + (−0.285 + 0.329i)10-s + (−0.608 − 0.391i)11-s + (0.242 + 0.156i)12-s + (−1.05 + 1.21i)13-s + (−0.256 − 0.0752i)14-s + (−0.0505 − 0.351i)15-s + (−0.163 − 0.188i)16-s + (0.343 + 0.751i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.659 - 0.751i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.659 - 0.751i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.284571 + 0.628639i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.284571 + 0.628639i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.841 + 0.540i)T \) |
| 3 | \( 1 + (0.142 - 0.989i)T \) |
| 7 | \( 1 + (0.654 + 0.755i)T \) |
| 23 | \( 1 + (-0.905 - 4.70i)T \) |
| good | 5 | \( 1 + (1.32 - 0.387i)T + (4.20 - 2.70i)T^{2} \) |
| 11 | \( 1 + (2.01 + 1.29i)T + (4.56 + 10.0i)T^{2} \) |
| 13 | \( 1 + (3.79 - 4.38i)T + (-1.85 - 12.8i)T^{2} \) |
| 17 | \( 1 + (-1.41 - 3.09i)T + (-11.1 + 12.8i)T^{2} \) |
| 19 | \( 1 + (1.16 - 2.55i)T + (-12.4 - 14.3i)T^{2} \) |
| 29 | \( 1 + (-0.563 - 1.23i)T + (-18.9 + 21.9i)T^{2} \) |
| 31 | \( 1 + (-0.105 - 0.735i)T + (-29.7 + 8.73i)T^{2} \) |
| 37 | \( 1 + (9.91 + 2.91i)T + (31.1 + 20.0i)T^{2} \) |
| 41 | \( 1 + (3.21 - 0.944i)T + (34.4 - 22.1i)T^{2} \) |
| 43 | \( 1 + (-1.33 + 9.28i)T + (-41.2 - 12.1i)T^{2} \) |
| 47 | \( 1 + 0.374T + 47T^{2} \) |
| 53 | \( 1 + (-1.39 - 1.60i)T + (-7.54 + 52.4i)T^{2} \) |
| 59 | \( 1 + (6.18 - 7.14i)T + (-8.39 - 58.3i)T^{2} \) |
| 61 | \( 1 + (0.465 + 3.23i)T + (-58.5 + 17.1i)T^{2} \) |
| 67 | \( 1 + (4.95 - 3.18i)T + (27.8 - 60.9i)T^{2} \) |
| 71 | \( 1 + (-3.52 + 2.26i)T + (29.4 - 64.5i)T^{2} \) |
| 73 | \( 1 + (-4.36 + 9.55i)T + (-47.8 - 55.1i)T^{2} \) |
| 79 | \( 1 + (3.13 - 3.61i)T + (-11.2 - 78.1i)T^{2} \) |
| 83 | \( 1 + (-15.5 - 4.56i)T + (69.8 + 44.8i)T^{2} \) |
| 89 | \( 1 + (1.01 - 7.05i)T + (-85.3 - 25.0i)T^{2} \) |
| 97 | \( 1 + (-0.383 + 0.112i)T + (81.6 - 52.4i)T^{2} \) |
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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
−10.45961016527101561951715767433, −9.709545248670210454652041296039, −8.817361767068746817902088074106, −7.68868165351428988443001205268, −6.91678675142292279398133553578, −5.79119222347938244402878388832, −4.95640646455497039602669987983, −3.94164454984658717573619140227, −3.34748150830748608511232882301, −1.93643669308534396066227545727,
0.24531912136854663618429847936, 2.39970590268046826272679231321, 3.20917763435855779544802874711, 4.67926276300007620229970551284, 5.23778923916708643853461848801, 6.32128699915769334849016153384, 7.22850089574889860954026283011, 7.83386753392050043128086438547, 8.549904921528947556990544042026, 9.752841644862960952004697244826
