| L(s) = 1 | + (−0.142 − 0.989i)2-s + (−0.415 + 0.909i)3-s + (−0.959 + 0.281i)4-s + (0.569 − 0.657i)5-s + (0.959 + 0.281i)6-s + (0.841 − 0.540i)7-s + (0.415 + 0.909i)8-s + (−0.654 − 0.755i)9-s + (−0.732 − 0.470i)10-s + (−0.719 + 5.00i)11-s + (0.142 − 0.989i)12-s + (−1.66 − 1.07i)13-s + (−0.654 − 0.755i)14-s + (0.361 + 0.791i)15-s + (0.841 − 0.540i)16-s + (6.13 + 1.79i)17-s + ⋯ |
| L(s) = 1 | + (−0.100 − 0.699i)2-s + (−0.239 + 0.525i)3-s + (−0.479 + 0.140i)4-s + (0.254 − 0.294i)5-s + (0.391 + 0.115i)6-s + (0.317 − 0.204i)7-s + (0.146 + 0.321i)8-s + (−0.218 − 0.251i)9-s + (−0.231 − 0.148i)10-s + (−0.217 + 1.50i)11-s + (0.0410 − 0.285i)12-s + (−0.461 − 0.296i)13-s + (−0.175 − 0.201i)14-s + (0.0933 + 0.204i)15-s + (0.210 − 0.135i)16-s + (1.48 + 0.436i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.921 - 0.389i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.921 - 0.389i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.26956 + 0.257300i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.26956 + 0.257300i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.142 + 0.989i)T \) |
| 3 | \( 1 + (0.415 - 0.909i)T \) |
| 7 | \( 1 + (-0.841 + 0.540i)T \) |
| 23 | \( 1 + (1.38 + 4.59i)T \) |
| good | 5 | \( 1 + (-0.569 + 0.657i)T + (-0.711 - 4.94i)T^{2} \) |
| 11 | \( 1 + (0.719 - 5.00i)T + (-10.5 - 3.09i)T^{2} \) |
| 13 | \( 1 + (1.66 + 1.07i)T + (5.40 + 11.8i)T^{2} \) |
| 17 | \( 1 + (-6.13 - 1.79i)T + (14.3 + 9.19i)T^{2} \) |
| 19 | \( 1 + (1.70 - 0.499i)T + (15.9 - 10.2i)T^{2} \) |
| 29 | \( 1 + (-0.574 - 0.168i)T + (24.3 + 15.6i)T^{2} \) |
| 31 | \( 1 + (-3.10 - 6.78i)T + (-20.3 + 23.4i)T^{2} \) |
| 37 | \( 1 + (-3.99 - 4.61i)T + (-5.26 + 36.6i)T^{2} \) |
| 41 | \( 1 + (0.129 - 0.149i)T + (-5.83 - 40.5i)T^{2} \) |
| 43 | \( 1 + (2.24 - 4.92i)T + (-28.1 - 32.4i)T^{2} \) |
| 47 | \( 1 + 1.50T + 47T^{2} \) |
| 53 | \( 1 + (-10.8 + 6.97i)T + (22.0 - 48.2i)T^{2} \) |
| 59 | \( 1 + (3.92 + 2.52i)T + (24.5 + 53.6i)T^{2} \) |
| 61 | \( 1 + (-3.87 - 8.47i)T + (-39.9 + 46.1i)T^{2} \) |
| 67 | \( 1 + (-2.06 - 14.3i)T + (-64.2 + 18.8i)T^{2} \) |
| 71 | \( 1 + (0.906 + 6.30i)T + (-68.1 + 20.0i)T^{2} \) |
| 73 | \( 1 + (-13.3 + 3.92i)T + (61.4 - 39.4i)T^{2} \) |
| 79 | \( 1 + (-7.35 - 4.72i)T + (32.8 + 71.8i)T^{2} \) |
| 83 | \( 1 + (-8.19 - 9.45i)T + (-11.8 + 82.1i)T^{2} \) |
| 89 | \( 1 + (1.96 - 4.31i)T + (-58.2 - 67.2i)T^{2} \) |
| 97 | \( 1 + (-1.94 + 2.24i)T + (-13.8 - 96.0i)T^{2} \) |
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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
−10.07111645177187123153378929248, −9.644832341556731389579645319067, −8.515915772757273291394933500552, −7.75912063847489820267656178195, −6.67859062564305390967997089111, −5.33872342775133275311255630675, −4.79226057120732272838512582349, −3.82241877208891165973005167961, −2.57427878315443832569445392646, −1.29630732988455481290383862139,
0.73319652183511758161702249222, 2.40299536078908705867112776663, 3.66485018574449826980157531254, 5.07734856157621714773818648123, 5.82437087944753722379118416008, 6.42522789633404336881507958770, 7.59163406694992864118104846525, 8.017783869828286402656977431517, 8.991139279788203606847758809069, 9.874694726176148823583052774020
