| L(s) = 1 | + (−0.741 − 1.20i)2-s + (−0.635 + 1.61i)3-s + (−0.901 + 1.78i)4-s + (−1.65 − 1.15i)5-s + (2.41 − 0.428i)6-s + (3.66 + 2.11i)7-s + (2.81 − 0.236i)8-s + (−2.19 − 2.04i)9-s + (−0.169 + 2.85i)10-s + (3.98 + 1.06i)11-s + (−2.30 − 2.58i)12-s + (−1.59 − 0.744i)13-s + (−0.167 − 5.99i)14-s + (2.92 − 1.93i)15-s + (−2.37 − 3.21i)16-s + (−0.339 − 0.404i)17-s + ⋯ |
| L(s) = 1 | + (−0.523 − 0.851i)2-s + (−0.366 + 0.930i)3-s + (−0.450 + 0.892i)4-s + (−0.740 − 0.518i)5-s + (0.984 − 0.174i)6-s + (1.38 + 0.800i)7-s + (0.996 − 0.0836i)8-s + (−0.730 − 0.682i)9-s + (−0.0536 + 0.902i)10-s + (1.20 + 0.322i)11-s + (−0.664 − 0.746i)12-s + (−0.442 − 0.206i)13-s + (−0.0447 − 1.60i)14-s + (0.754 − 0.498i)15-s + (−0.593 − 0.804i)16-s + (−0.0823 − 0.0981i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.863 - 0.504i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.863 - 0.504i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.960742 + 0.259853i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.960742 + 0.259853i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.741 + 1.20i)T \) |
| 3 | \( 1 + (0.635 - 1.61i)T \) |
| 19 | \( 1 + (0.601 + 4.31i)T \) |
| good | 5 | \( 1 + (1.65 + 1.15i)T + (1.71 + 4.69i)T^{2} \) |
| 7 | \( 1 + (-3.66 - 2.11i)T + (3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (-3.98 - 1.06i)T + (9.52 + 5.5i)T^{2} \) |
| 13 | \( 1 + (1.59 + 0.744i)T + (8.35 + 9.95i)T^{2} \) |
| 17 | \( 1 + (0.339 + 0.404i)T + (-2.95 + 16.7i)T^{2} \) |
| 23 | \( 1 + (1.23 - 6.98i)T + (-21.6 - 7.86i)T^{2} \) |
| 29 | \( 1 + (-2.99 + 0.262i)T + (28.5 - 5.03i)T^{2} \) |
| 31 | \( 1 + (-6.14 - 3.55i)T + (15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (-3.28 + 3.28i)T - 37iT^{2} \) |
| 41 | \( 1 + (3.54 - 9.73i)T + (-31.4 - 26.3i)T^{2} \) |
| 43 | \( 1 + (1.19 - 1.69i)T + (-14.7 - 40.4i)T^{2} \) |
| 47 | \( 1 + (3.06 - 3.65i)T + (-8.16 - 46.2i)T^{2} \) |
| 53 | \( 1 + (-9.97 + 6.98i)T + (18.1 - 49.8i)T^{2} \) |
| 59 | \( 1 + (-13.0 - 1.13i)T + (58.1 + 10.2i)T^{2} \) |
| 61 | \( 1 + (0.519 + 0.741i)T + (-20.8 + 57.3i)T^{2} \) |
| 67 | \( 1 + (-1.31 - 15.0i)T + (-65.9 + 11.6i)T^{2} \) |
| 71 | \( 1 + (-3.87 + 0.682i)T + (66.7 - 24.2i)T^{2} \) |
| 73 | \( 1 + (4.58 - 12.5i)T + (-55.9 - 46.9i)T^{2} \) |
| 79 | \( 1 + (0.863 - 2.37i)T + (-60.5 - 50.7i)T^{2} \) |
| 83 | \( 1 + (11.7 - 3.14i)T + (71.8 - 41.5i)T^{2} \) |
| 89 | \( 1 + (-1.36 - 3.75i)T + (-68.1 + 57.2i)T^{2} \) |
| 97 | \( 1 + (-6.02 - 7.18i)T + (-16.8 + 95.5i)T^{2} \) |
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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
−10.07098864973397857329889937111, −9.429040182538700166033892909675, −8.548471668131212393841820913105, −8.200678880374374325173567148212, −6.92345479240662387592544986561, −5.35255254747111846978403845622, −4.60008327747425080039569331068, −4.00585199510613684196977666399, −2.65304783773243611103217151842, −1.15073965709933550536787244978,
0.75512159134362373155146849273, 1.92665690231911821870944597057, 3.99165883657626402443535040126, 4.82556461980893530533522423078, 6.04841941273439613070613641238, 6.81844121718591038498742898066, 7.44328916045484097859290149179, 8.178924151110787174259865059349, 8.672657908923908754250586046556, 10.17519734160954288795301366820
