| L(s) = 1 | + (−0.142 − 0.989i)2-s + (−0.909 − 0.415i)3-s + (−0.959 + 0.281i)4-s + (−1.30 + 1.50i)5-s + (−0.281 + 0.959i)6-s + (2.10 − 1.60i)7-s + (0.415 + 0.909i)8-s + (0.654 + 0.755i)9-s + (1.67 + 1.07i)10-s + (−5.59 − 0.804i)11-s + (0.989 + 0.142i)12-s + (2.36 − 3.68i)13-s + (−1.88 − 1.85i)14-s + (1.81 − 0.828i)15-s + (0.841 − 0.540i)16-s + (6.51 + 1.91i)17-s + ⋯ |
| L(s) = 1 | + (−0.100 − 0.699i)2-s + (−0.525 − 0.239i)3-s + (−0.479 + 0.140i)4-s + (−0.583 + 0.673i)5-s + (−0.115 + 0.391i)6-s + (0.795 − 0.605i)7-s + (0.146 + 0.321i)8-s + (0.218 + 0.251i)9-s + (0.530 + 0.340i)10-s + (−1.68 − 0.242i)11-s + (0.285 + 0.0410i)12-s + (0.656 − 1.02i)13-s + (−0.504 − 0.495i)14-s + (0.468 − 0.213i)15-s + (0.210 − 0.135i)16-s + (1.58 + 0.464i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.847 - 0.530i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.847 - 0.530i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.0641231 + 0.223413i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0641231 + 0.223413i\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + (0.909 + 0.415i)T \) |
| 7 | \( 1 + (-2.10 + 1.60i)T \) |
| 23 | \( 1 + (4.47 + 1.71i)T \) |
| good | 5 | \( 1 + (1.30 - 1.50i)T + (-0.711 - 4.94i)T^{2} \) |
| 11 | \( 1 + (5.59 + 0.804i)T + (10.5 + 3.09i)T^{2} \) |
| 13 | \( 1 + (-2.36 + 3.68i)T + (-5.40 - 11.8i)T^{2} \) |
| 17 | \( 1 + (-6.51 - 1.91i)T + (14.3 + 9.19i)T^{2} \) |
| 19 | \( 1 + (0.721 - 0.211i)T + (15.9 - 10.2i)T^{2} \) |
| 29 | \( 1 + (8.03 + 2.35i)T + (24.3 + 15.6i)T^{2} \) |
| 31 | \( 1 + (6.73 - 3.07i)T + (20.3 - 23.4i)T^{2} \) |
| 37 | \( 1 + (-1.80 + 1.56i)T + (5.26 - 36.6i)T^{2} \) |
| 41 | \( 1 + (8.72 + 7.55i)T + (5.83 + 40.5i)T^{2} \) |
| 43 | \( 1 + (8.11 + 3.70i)T + (28.1 + 32.4i)T^{2} \) |
| 47 | \( 1 - 8.92iT - 47T^{2} \) |
| 53 | \( 1 + (-2.91 - 4.52i)T + (-22.0 + 48.2i)T^{2} \) |
| 59 | \( 1 + (5.04 - 7.85i)T + (-24.5 - 53.6i)T^{2} \) |
| 61 | \( 1 + (2.86 + 6.26i)T + (-39.9 + 46.1i)T^{2} \) |
| 67 | \( 1 + (6.35 - 0.914i)T + (64.2 - 18.8i)T^{2} \) |
| 71 | \( 1 + (0.385 + 2.68i)T + (-68.1 + 20.0i)T^{2} \) |
| 73 | \( 1 + (0.959 + 3.26i)T + (-61.4 + 39.4i)T^{2} \) |
| 79 | \( 1 + (9.34 - 14.5i)T + (-32.8 - 71.8i)T^{2} \) |
| 83 | \( 1 + (-2.98 - 3.44i)T + (-11.8 + 82.1i)T^{2} \) |
| 89 | \( 1 + (-2.71 + 5.95i)T + (-58.2 - 67.2i)T^{2} \) |
| 97 | \( 1 + (-2.35 + 2.72i)T + (-13.8 - 96.0i)T^{2} \) |
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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
−10.02652246230700952368855844270, −8.471337687808452102744354744521, −7.73709373827108395611863518357, −7.42451882188837922649190275071, −5.76904351323968953762853971989, −5.25075274467734577558280722228, −3.88484948315888760403941350487, −3.11977897868836997520296324303, −1.66297971830108954053928120111, −0.12030672739486755943660699594,
1.72783541092415766566123026025, 3.57525880939231566678980103409, 4.74982023682652371055730511952, 5.26025307086165462145333326923, 6.01491188291541085720686917467, 7.37270715446841339734827789080, 7.961550109038530458678762000086, 8.633729619142075331804579367839, 9.604862130395522113709804538220, 10.39104949193720960144362679512
