| L(s) = 1 | + (−0.793 − 1.17i)2-s + (1.68 + 0.898i)3-s + (−0.742 + 1.85i)4-s + (−1.56 + 1.91i)5-s + (−0.281 − 2.68i)6-s + (3.63 + 2.43i)7-s + (2.76 − 0.603i)8-s + (0.353 + 0.529i)9-s + (3.48 + 0.321i)10-s + (−4.39 − 1.33i)11-s + (−2.91 + 2.45i)12-s + (1.74 − 1.42i)13-s + (−0.0388 − 6.19i)14-s + (−4.35 + 1.80i)15-s + (−2.89 − 2.75i)16-s + (3.75 + 1.55i)17-s + ⋯ |
| L(s) = 1 | + (−0.560 − 0.827i)2-s + (0.970 + 0.518i)3-s + (−0.371 + 0.928i)4-s + (−0.701 + 0.854i)5-s + (−0.114 − 1.09i)6-s + (1.37 + 0.919i)7-s + (0.976 − 0.213i)8-s + (0.117 + 0.176i)9-s + (1.10 + 0.101i)10-s + (−1.32 − 0.401i)11-s + (−0.842 + 0.709i)12-s + (0.483 − 0.396i)13-s + (−0.0103 − 1.65i)14-s + (−1.12 + 0.466i)15-s + (−0.724 − 0.689i)16-s + (0.909 + 0.376i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 128 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.981 - 0.189i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 128 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.981 - 0.189i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.02299 + 0.0980705i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.02299 + 0.0980705i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.793 + 1.17i)T \) |
| good | 3 | \( 1 + (-1.68 - 0.898i)T + (1.66 + 2.49i)T^{2} \) |
| 5 | \( 1 + (1.56 - 1.91i)T + (-0.975 - 4.90i)T^{2} \) |
| 7 | \( 1 + (-3.63 - 2.43i)T + (2.67 + 6.46i)T^{2} \) |
| 11 | \( 1 + (4.39 + 1.33i)T + (9.14 + 6.11i)T^{2} \) |
| 13 | \( 1 + (-1.74 + 1.42i)T + (2.53 - 12.7i)T^{2} \) |
| 17 | \( 1 + (-3.75 - 1.55i)T + (12.0 + 12.0i)T^{2} \) |
| 19 | \( 1 + (-0.804 + 8.16i)T + (-18.6 - 3.70i)T^{2} \) |
| 23 | \( 1 + (1.74 + 0.347i)T + (21.2 + 8.80i)T^{2} \) |
| 29 | \( 1 + (-0.598 - 1.97i)T + (-24.1 + 16.1i)T^{2} \) |
| 31 | \( 1 + (3.81 - 3.81i)T - 31iT^{2} \) |
| 37 | \( 1 + (1.04 - 0.102i)T + (36.2 - 7.21i)T^{2} \) |
| 41 | \( 1 + (-0.711 + 3.57i)T + (-37.8 - 15.6i)T^{2} \) |
| 43 | \( 1 + (0.793 - 0.423i)T + (23.8 - 35.7i)T^{2} \) |
| 47 | \( 1 + (-1.43 + 3.46i)T + (-33.2 - 33.2i)T^{2} \) |
| 53 | \( 1 + (-1.79 + 5.91i)T + (-44.0 - 29.4i)T^{2} \) |
| 59 | \( 1 + (4.26 + 3.49i)T + (11.5 + 57.8i)T^{2} \) |
| 61 | \( 1 + (2.89 - 5.41i)T + (-33.8 - 50.7i)T^{2} \) |
| 67 | \( 1 + (2.61 - 4.89i)T + (-37.2 - 55.7i)T^{2} \) |
| 71 | \( 1 + (-0.916 + 1.37i)T + (-27.1 - 65.5i)T^{2} \) |
| 73 | \( 1 + (-6.52 + 4.36i)T + (27.9 - 67.4i)T^{2} \) |
| 79 | \( 1 + (4.05 + 9.80i)T + (-55.8 + 55.8i)T^{2} \) |
| 83 | \( 1 + (0.630 + 0.0621i)T + (81.4 + 16.1i)T^{2} \) |
| 89 | \( 1 + (-3.26 + 0.649i)T + (82.2 - 34.0i)T^{2} \) |
| 97 | \( 1 + (12.6 - 12.6i)T - 97iT^{2} \) |
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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
−13.38296094840089922269989269265, −12.03027316090919291868381780690, −11.11111147183805602827016805586, −10.47816478386126404860251873630, −9.006578995458521407068255385024, −8.301739592640086702681419332310, −7.54490164628953767390668762033, −5.09211553610704047339388490087, −3.45210484051890464526484260178, −2.54110431857810722440460355564,
1.54360325812182180933004949612, 4.24003204819980336188134041832, 5.43741241230721573118911457962, 7.66199479188589657737530140411, 7.76869991972290422110601982939, 8.507370103133028687115291104304, 9.959551090229819163050929908092, 11.08739180895677751576310876865, 12.51954024286557126994115700370, 13.76355045996499708643853754561
