| L(s) = 1 | + (0.142 + 0.989i)2-s + (−0.415 + 0.909i)3-s + (−0.959 + 0.281i)4-s + (−1.94 + 2.24i)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 + (−2.49 − 1.60i)10-s + (0.361 − 2.51i)11-s + (0.142 − 0.989i)12-s + (−5.77 − 3.71i)13-s + (−0.654 − 0.755i)14-s + (−1.23 − 2.70i)15-s + (0.841 − 0.540i)16-s + (3.79 + 1.11i)17-s + ⋯ |
| L(s) = 1 | + (0.100 + 0.699i)2-s + (−0.239 + 0.525i)3-s + (−0.479 + 0.140i)4-s + (−0.869 + 1.00i)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 + (−0.790 − 0.507i)10-s + (0.109 − 0.758i)11-s + (0.0410 − 0.285i)12-s + (−1.60 − 1.02i)13-s + (−0.175 − 0.201i)14-s + (−0.318 − 0.697i)15-s + (0.210 − 0.135i)16-s + (0.920 + 0.270i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.703 + 0.710i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.703 + 0.710i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.260491 - 0.108606i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.260491 - 0.108606i\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + (-3.74 + 2.99i)T \) |
| good | 5 | \( 1 + (1.94 - 2.24i)T + (-0.711 - 4.94i)T^{2} \) |
| 11 | \( 1 + (-0.361 + 2.51i)T + (-10.5 - 3.09i)T^{2} \) |
| 13 | \( 1 + (5.77 + 3.71i)T + (5.40 + 11.8i)T^{2} \) |
| 17 | \( 1 + (-3.79 - 1.11i)T + (14.3 + 9.19i)T^{2} \) |
| 19 | \( 1 + (0.341 - 0.100i)T + (15.9 - 10.2i)T^{2} \) |
| 29 | \( 1 + (5.47 + 1.60i)T + (24.3 + 15.6i)T^{2} \) |
| 31 | \( 1 + (-3.86 - 8.45i)T + (-20.3 + 23.4i)T^{2} \) |
| 37 | \( 1 + (2.60 + 3.00i)T + (-5.26 + 36.6i)T^{2} \) |
| 41 | \( 1 + (-1.43 + 1.65i)T + (-5.83 - 40.5i)T^{2} \) |
| 43 | \( 1 + (1.87 - 4.11i)T + (-28.1 - 32.4i)T^{2} \) |
| 47 | \( 1 + 7.94T + 47T^{2} \) |
| 53 | \( 1 + (-10.3 + 6.66i)T + (22.0 - 48.2i)T^{2} \) |
| 59 | \( 1 + (5.59 + 3.59i)T + (24.5 + 53.6i)T^{2} \) |
| 61 | \( 1 + (-0.303 - 0.665i)T + (-39.9 + 46.1i)T^{2} \) |
| 67 | \( 1 + (0.906 + 6.30i)T + (-64.2 + 18.8i)T^{2} \) |
| 71 | \( 1 + (1.57 + 10.9i)T + (-68.1 + 20.0i)T^{2} \) |
| 73 | \( 1 + (-2.57 + 0.756i)T + (61.4 - 39.4i)T^{2} \) |
| 79 | \( 1 + (13.8 + 8.92i)T + (32.8 + 71.8i)T^{2} \) |
| 83 | \( 1 + (6.34 + 7.32i)T + (-11.8 + 82.1i)T^{2} \) |
| 89 | \( 1 + (-3.81 + 8.35i)T + (-58.2 - 67.2i)T^{2} \) |
| 97 | \( 1 + (6.69 - 7.72i)T + (-13.8 - 96.0i)T^{2} \) |
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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
−10.06156546613963202367474614524, −9.003989291139736095326284666467, −8.040994066070368632476822970520, −7.36804303323051438798581213610, −6.56475535303510286334633301193, −5.57631388347087640563715570982, −4.78416924887711669526573365611, −3.47848342851728486583546211521, −2.97582166271242539182487615680, −0.14157340313262724666983133414,
1.27827320724743924822280510612, 2.56540555133614828188245399553, 3.97585550604194067205726735046, 4.70108693202045027328838744076, 5.51745873693415581678534408069, 7.03045900552377780120716329574, 7.51248105011570797358363379489, 8.517552165864939772226669595660, 9.523672766618908769804922409580, 9.912612562123857824049756566847
