| L(s) = 1 | + (0.654 + 0.755i)2-s + (−0.841 − 0.540i)3-s + (−0.142 + 0.989i)4-s + (0.702 − 1.53i)5-s + (−0.142 − 0.989i)6-s + (−0.959 − 0.281i)7-s + (−0.841 + 0.540i)8-s + (0.415 + 0.909i)9-s + (1.62 − 0.476i)10-s + (−1.30 + 1.50i)11-s + (0.654 − 0.755i)12-s + (−6.62 + 1.94i)13-s + (−0.415 − 0.909i)14-s + (−1.42 + 0.913i)15-s + (−0.959 − 0.281i)16-s + (0.346 + 2.41i)17-s + ⋯ |
| L(s) = 1 | + (0.463 + 0.534i)2-s + (−0.485 − 0.312i)3-s + (−0.0711 + 0.494i)4-s + (0.314 − 0.687i)5-s + (−0.0580 − 0.404i)6-s + (−0.362 − 0.106i)7-s + (−0.297 + 0.191i)8-s + (0.138 + 0.303i)9-s + (0.512 − 0.150i)10-s + (−0.392 + 0.452i)11-s + (0.189 − 0.218i)12-s + (−1.83 + 0.539i)13-s + (−0.111 − 0.243i)14-s + (−0.367 + 0.235i)15-s + (−0.239 − 0.0704i)16-s + (0.0841 + 0.585i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.678 - 0.734i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.678 - 0.734i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.376645 + 0.860243i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.376645 + 0.860243i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.654 - 0.755i)T \) |
| 3 | \( 1 + (0.841 + 0.540i)T \) |
| 7 | \( 1 + (0.959 + 0.281i)T \) |
| 23 | \( 1 + (-2.67 - 3.98i)T \) |
| good | 5 | \( 1 + (-0.702 + 1.53i)T + (-3.27 - 3.77i)T^{2} \) |
| 11 | \( 1 + (1.30 - 1.50i)T + (-1.56 - 10.8i)T^{2} \) |
| 13 | \( 1 + (6.62 - 1.94i)T + (10.9 - 7.02i)T^{2} \) |
| 17 | \( 1 + (-0.346 - 2.41i)T + (-16.3 + 4.78i)T^{2} \) |
| 19 | \( 1 + (0.435 - 3.03i)T + (-18.2 - 5.35i)T^{2} \) |
| 29 | \( 1 + (-1.18 - 8.21i)T + (-27.8 + 8.17i)T^{2} \) |
| 31 | \( 1 + (-6.32 + 4.06i)T + (12.8 - 28.1i)T^{2} \) |
| 37 | \( 1 + (-1.01 - 2.21i)T + (-24.2 + 27.9i)T^{2} \) |
| 41 | \( 1 + (0.350 - 0.766i)T + (-26.8 - 30.9i)T^{2} \) |
| 43 | \( 1 + (-0.566 - 0.363i)T + (17.8 + 39.1i)T^{2} \) |
| 47 | \( 1 + 11.4T + 47T^{2} \) |
| 53 | \( 1 + (9.12 + 2.68i)T + (44.5 + 28.6i)T^{2} \) |
| 59 | \( 1 + (-2.40 + 0.707i)T + (49.6 - 31.8i)T^{2} \) |
| 61 | \( 1 + (4.82 - 3.09i)T + (25.3 - 55.4i)T^{2} \) |
| 67 | \( 1 + (2.44 + 2.81i)T + (-9.53 + 66.3i)T^{2} \) |
| 71 | \( 1 + (0.623 + 0.719i)T + (-10.1 + 70.2i)T^{2} \) |
| 73 | \( 1 + (1.94 - 13.5i)T + (-70.0 - 20.5i)T^{2} \) |
| 79 | \( 1 + (1.81 - 0.531i)T + (66.4 - 42.7i)T^{2} \) |
| 83 | \( 1 + (6.72 + 14.7i)T + (-54.3 + 62.7i)T^{2} \) |
| 89 | \( 1 + (3.41 + 2.19i)T + (36.9 + 80.9i)T^{2} \) |
| 97 | \( 1 + (-3.62 + 7.93i)T + (-63.5 - 73.3i)T^{2} \) |
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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
−10.09967676435266117625911519082, −9.595630362617201384148867413467, −8.524042357778863450767654728749, −7.55154300841466220713877926868, −6.94023543226666899546196454786, −5.99014829505572603269615851959, −5.04109880415048973606221452615, −4.58155322276411090885790330394, −3.07122311308658130662246173428, −1.66919830330494640212653316060,
0.37900142371983349162678241694, 2.56455402004156373377474665105, 2.98244452733266072928330582741, 4.56634182031427658048161420703, 5.12558441855432247253172688350, 6.22823952165052154264942997247, 6.88099051954095013983401185664, 7.982734461197238350805805154652, 9.278289584245356896863859414168, 9.997961295928409674532151195752
