| L(s) = 1 | + (0.654 − 0.755i)2-s + (0.841 − 0.540i)3-s + (−0.142 − 0.989i)4-s + (0.817 + 1.78i)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 + (1.88 + 0.554i)10-s + (0.898 + 1.03i)11-s + (−0.654 − 0.755i)12-s + (5.03 + 1.47i)13-s + (−0.415 + 0.909i)14-s + (1.65 + 1.06i)15-s + (−0.959 + 0.281i)16-s + (0.490 − 3.41i)17-s + ⋯ |
| L(s) = 1 | + (0.463 − 0.534i)2-s + (0.485 − 0.312i)3-s + (−0.0711 − 0.494i)4-s + (0.365 + 0.800i)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 + (0.596 + 0.175i)10-s + (0.271 + 0.312i)11-s + (−0.189 − 0.218i)12-s + (1.39 + 0.409i)13-s + (−0.111 + 0.243i)14-s + (0.427 + 0.274i)15-s + (−0.239 + 0.0704i)16-s + (0.118 − 0.827i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.739 + 0.673i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.739 + 0.673i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.44067 - 0.945242i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.44067 - 0.945242i\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + (-3.99 - 2.64i)T \) |
| good | 5 | \( 1 + (-0.817 - 1.78i)T + (-3.27 + 3.77i)T^{2} \) |
| 11 | \( 1 + (-0.898 - 1.03i)T + (-1.56 + 10.8i)T^{2} \) |
| 13 | \( 1 + (-5.03 - 1.47i)T + (10.9 + 7.02i)T^{2} \) |
| 17 | \( 1 + (-0.490 + 3.41i)T + (-16.3 - 4.78i)T^{2} \) |
| 19 | \( 1 + (0.545 + 3.79i)T + (-18.2 + 5.35i)T^{2} \) |
| 29 | \( 1 + (0.413 - 2.87i)T + (-27.8 - 8.17i)T^{2} \) |
| 31 | \( 1 + (-6.33 - 4.07i)T + (12.8 + 28.1i)T^{2} \) |
| 37 | \( 1 + (-2.60 + 5.70i)T + (-24.2 - 27.9i)T^{2} \) |
| 41 | \( 1 + (0.924 + 2.02i)T + (-26.8 + 30.9i)T^{2} \) |
| 43 | \( 1 + (3.99 - 2.56i)T + (17.8 - 39.1i)T^{2} \) |
| 47 | \( 1 + 6.48T + 47T^{2} \) |
| 53 | \( 1 + (-9.57 + 2.81i)T + (44.5 - 28.6i)T^{2} \) |
| 59 | \( 1 + (4.98 + 1.46i)T + (49.6 + 31.8i)T^{2} \) |
| 61 | \( 1 + (6.67 + 4.29i)T + (25.3 + 55.4i)T^{2} \) |
| 67 | \( 1 + (10.0 - 11.5i)T + (-9.53 - 66.3i)T^{2} \) |
| 71 | \( 1 + (2.16 - 2.49i)T + (-10.1 - 70.2i)T^{2} \) |
| 73 | \( 1 + (-1.08 - 7.52i)T + (-70.0 + 20.5i)T^{2} \) |
| 79 | \( 1 + (-3.45 - 1.01i)T + (66.4 + 42.7i)T^{2} \) |
| 83 | \( 1 + (3.28 - 7.19i)T + (-54.3 - 62.7i)T^{2} \) |
| 89 | \( 1 + (9.33 - 5.99i)T + (36.9 - 80.9i)T^{2} \) |
| 97 | \( 1 + (7.70 + 16.8i)T + (-63.5 + 73.3i)T^{2} \) |
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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
−9.937856737515540485780403054478, −9.200518553371495432427365621438, −8.461862017685840079328279408525, −7.00209201232992921312399318165, −6.68689525236127975946921211932, −5.61776887676227110794724687389, −4.41330494687582270285537810045, −3.28620280318406598720068653769, −2.64181182550621425606749898586, −1.31548876609415637346250030222,
1.34791442380542821855832263092, 3.04779923232769956015805393418, 3.91646316531317049562855663090, 4.81343350920309104247978748328, 5.93527738672489240136451342939, 6.39602730201666107180271641418, 7.80315078608391624172926013449, 8.470513890512087093875241742620, 9.023450058738368659851801249934, 10.02039908371597630253647163412
